ON CONNECTEDNESS OF SETS IN THE REAL SPECTRA OF POLYNOMIAL RINGS

F. Lucas

Département de Mathématiques

Université d’Angers 2, bd Lavoisier

49045 Angers Cedex, France

J. J. Madden Department of Mathematics Louisiana State University at Baton Rouge, Baton Rouge, LA, USA

D. Schaub

Département de Mathématiques

Université d’Angers

2, bd Lavoisier 49045 Angers Cedex, France

M. Spivakovsky CNRS-Institut de Mathématiques de Toulouse Université Paul Sabatier 118, route de Narbonne 31062 Toulouse Cedex 9, France.

Abstract

Let R be a real closed field. The Pierce-Birkhoff conjecture says that any piecewise polynomial function f on R^{n} can be obtained from the polynomial ring R[x_{1}, …, x_{n}] by iterating the operations of maximum and minimum. The purpose of this paper is threefold. First, we state a new conjecture, called the Connectedness conjecture, which asserts, for every pair of points α, β ∈ Sper R[x_{1}, …, x_{n}], the existence of connected sets in the real spectrum of R[x_{1}, …, x_{n}], satisfying certain conditions. We prove that the Connectedness conjecture implies the Pierce-Birkhoff conjecture.

Secondly, we construct a class of connected sets in the real spectrum which, though not in itself enough for the proof of the Pierce-Birkhoff conjecture, is the first and simplest example of the sort of connected sets we really need, and which constitutes the first step in our program for a proof of the Pierce-Birkhoff conjecture in dimension greater than 2.

Thirdly, we apply these ideas to give two proofs of the Connectedness conjecture (and hence also of the Pierce–Birkhoff conjecture in the abstract formulation) in the special case when one of the two points α, β ∈ Sper R[x_{1}, …, x_{n}] is monomial. One of the proofs is elementary while the other consists in deducing the (monomial) Connectedness conjecture as an immediate corollary of the main connectedness theorem of this paper.

1 Introduction

All the rings in this paper will be commutative with 1. Throughout this paper, R will denote a real closed field and A the polynomial ring R[x_{1}, …, x_{n}], unless otherwise specified.

The Pierce-Birkhoff conjecture asserts that any piecewise-polynomial function f : R^{n} → R can be expressed as a maximum of minima of a finite family of polynomials (see below for the definitions and a precise statement of the conjecture). This paper is the first step in our program for a proof of the Pierce-Birkhoff conjecture in its full generality (the best results up to now are due to Louis Mahé [15], who proved the conjecture for n = 2, as well as some partial results for n = 3).

We start by stating the Pierce–Birkhoff conjecture in its original form as it was first stated by M. Henriksen and J. Isbell in the early nineteen sixties.

Definition 1.1 A function f : R^{n} → R is said to be piecewise polynomial if R^{n} can be covered by a finite collection of closed semi-algebraic sets P_{i} such that for each i there exists a polynomial f_{i} ∈ A satisfying f|_{P_{i}} = f_{i}|_{P_{i}}. Given a piecewise polynomial function f, we say that f is defined by r polynomials if r is the number of distinct polynomials among the f_{i} above.

Clearly, any piecewise polynomial function is continuous. Piecewise polynomial functions form a ring, containing A, which is denoted by P W (A).

On the other hand, one can consider the (lattice-ordered) ring of all the functions obtained from A by iterating the operations of sup and inf. Since applying the operations of sup and inf to polynomials produces functions which are piecewise polynomial, this ring is contained in P W (A) (the latter ring is closed under sup and inf). It is natural to ask whether the two rings coincide. The precise statement of the conjecture is:

Conjecture 1 (Pierce-Birkhoff) If f : R^{n} → R is in P W (A), then there exists a finite family of polynomials g_{ij} ∈ A such that f = sup_{i} inf_{j}(g_{ij}) (in other words, for all x ∈ R^{n},

f(x) = sup_{i} inf_{j}(g_{ij}(x))).

A few words about the history of the problem. A question suggestive of the present Pierce– Birkhoff conjecture appeared in the 1956 paper of Birkhoff and Pierce [4], where f-rings were first defined. The question seems to have been phrased somewhat carelessly, and the intended meaning of the text is not easy to understand.

According to Mel Henriksen, he and Isbell attempted to prove the conjecture as it is now formulated while working on f-rings in the early nineteen sixties [8]. Since they obtained no significant results about it, they never mentioned it explicitly in print.

In the early nineteen eighties, when real algebraic geometry was becoming active, Isbell described the question to several mathematicians at several meetings. Efroymson and Mahé took up the challenge and worked on it.

Isbell did not hesitate to read the 1956 question posed by Birkhoff and Pierce quite generously. He seems to be the one responsible for attaching the name “Pierce–Birkhoff conjecture” to the problem. He believed that Birkhoff and Pierce intended to ask the very question we now call the Pierce–Birkhoff conjecture.

It is not clear how or why the names Pierce and Birkhoff came to appear in reverse alphabetical order, rather than the alphabetical order that was used in their 1956 paper. In light of the history, it might be more accurate to call the question the “Pierce-Birkhoff-Isbell-Henriksen conjecture”.

In 1989 J.J. Madden [13] reformulated this conjecture in terms of the real spectrum of A and separating ideals. We now recall Madden’s formulation together with the relevant definitions.

Let B be a ring. A point α in the real spectrum of B is, by definition, the data of a prime ideal 𝔭 of B, and a total ordering ≤ of the quotient ring B/𝔭, or, equivalently, of the field of fractions of B/𝔭. Another way of defining the point α is as a homomorphism from B to a real closed field, where two homomorphisms are identified if they have the same kernel 𝔭 and induce the same total ordering on B/𝔭.

The ideal 𝔭 is called the support of α and denoted by 𝔭_{α}, the quotient ring B/𝔭_{α} by B[α], its field of fractions by B(α) and the real closure of B(α) by k(α). The total ordering of B(α) is denoted by ≤_{α}. Sometimes we write α = (𝔭_{α}, ≤_{α}).

Definition 1.2 The real spectrum of B, denoted by Sper B, is the collection of all pairs α = (𝔭_{α}, ≤_{α}), where 𝔭_{α} is a prime ideal of B and ≤_{α} is a total ordering of B/𝔭_{α}.

The real spectrum Sper B is endowed with two natural topologies. The first one, called the spectral (or Harrison) topology, has basic open sets of the form

U(f_{1}, …, f_{k}) = {α | f_{1}(α) > 0, …, f_{k}(α) > 0}

with f_{1}, ..., f_{k} ∈ B. Here and below, we commit the following standard abuse of notation: for an element f ∈ B, f(α) stands for the natural image of f in B[α] and the inequality f(α) > 0 really means f(α) >_{α} 0.

The second is the constructible topology whose basic open sets are of the form

V (f_{1}, …, f_{k}, g) = {α | f_{1}(α) > 0, …, f_{k}(α) > 0, g(α) = 0},

where f_{1}, ..., f_{n}, g ∈ B. Boolean combinations of sets of the form V (f_{1}, …, f_{n}, g) are called constructible sets of Sper B.

Proposition 1.1 ([5], Proposition 7.1.12, p.114) Let B be any ring, then Sper(B) is compact for the constructible topology.

Remark 1.1 Since the spectral topology is coarser than the constructible topology, Sper(B) is also compact for the spectral topology.

Denote by Maxr(A) the set of points α ∈ Sper(A) such that 𝔭_{α} is a maximal ideal of A. We view Maxr(A) as a topological subspace of Sper(A) with the spectral (respectively, constructible) topology. We may naturally identify R^{n} with Maxr(A): a point (a_{1}, …, a_{n}) ∈ R^{n} corresponds to the point α = (𝔭_{α}, ≤_{α}) ∈ Sper(A), where 𝔭_{α} is the maximal ideal

𝔭_{α} = (x_{1} − a_{1}, …, x_{n} − a_{n})

and ≤_{α} is the unique order on R. The spectral topology on Sper(A) induces the euclidean topology on R^{n}, while the constructible topology induces the topology of R^{n} whose base consists of the semi-algebraic sets. The injection R^{n} = Maxr(A) ↪ Sper(A) induces a one-to-one correspondence between the semi-algebraic sets of R^{n} and the constructible sets of Sper(A): the set S(f_{1}, …, f_{k}, g) = {(a_{1}, …, a_{n}) ∈ R^{n} | f_{1}(a_{1}, …, a_{n}) > 0, …, f_{k}(a_{1}, …, a_{n}) > 0, g(a_{1}, …, a_{n}) = 0} corresponds to the set

V (f_{1}, …, f_{k}, g) = {α | f_{1}(α) > 0, …, f_{k}(α) > 0, g(α) = 0}.

For a general semi-algebraic set S in R^{n}, the corresponding constructible set of Sper(A) will be denoted by S̃. Conversely, we sometimes start with a constructible set S̃ in Sper(A) and denote by S its semi-algebraic counterpart in R^{n}.

Notation: We will use the notation ∐ to denote the disjoint union of sets.

Recall Definition 2.4.2 and Proposition 7.5.1 of [5] (p. 31 and p. 130, respectively):

Definition 1.3 Let S be a semi-algebraic subset of R^{n}. The set S is called semi-algebraically connected if, for any decomposition S = S_{1} ∐ S_{2}, with S_{1}, S_{2} semi-algebraic and relatively closed in S for the euclidean topology, we have either S_{1} = S or S_{2} = S.

Proposition 1.2 A semi-algebraic set S in R^{n} is semi-algebraically connected if and only if S̃ is connected in the spectral topology. If S_{1}, …, S_{k} are the semi-algebraically connected components of S, then S̃_{1}, …, S̃_{k} are the connected components of S̃.

Warning: Semi-algebraic connectedness is the same as the usual connectedness if R = ℝ, but not in general. For a general real closed field R, usual connectedness in the euclidean topology is a very restrictive notion and will not be used in this paper.

Proposition 1.3 ([5], Theorems 2.4.4 and 7.5.1). Let A = R[x_{1}, …, x_{n}] and let X be a constructible subset of Sper(A). Then X can be written as

X = ∐_{i=1}^{r} X_{i}  (1)

where each X_{i} is constructible and connected in the spectral topology. In other words, X has finitely many connected components in the spectral topology, each of which is itself constructible.

Remark 1.2 Let B be a finitely generated R-algebra. Then Proposition 1.3 holds also for Sper A replaced with Sper B. Indeed, B is a homomorphic image of a polynomial ring of the form R[x_{1}, …, x_{n}], so Sper B is a closed subspace of Sper R[x_{1}, …, x_{n}]. Then a constructible subset X of Sper B is also constructible viewed as a subset of Sper R[x_{1}, …, x_{n}] and the decomposition (1) has the properties required in the Proposition regardless of whether one views the X_{i} as subsets of Sper B or of Sper R[x_{1}, …, x_{n}].

Next, we recall the notion of separating ideal, introduced by Madden in [13].

Definition 1.4 Let B be a ring. For α, β ∈ Sper B, the separating ideal of α and β, denoted by < α, β >, is the ideal of B generated by all the elements f ∈ B which change sign between α and β, that is, all the f such that f(α) ≥ 0 and f(β) ≤ 0.

We will need the following basic properties of the separating ideal, proved in [13]:

Proposition 1.4 Let the notation be as above. We have:

(1) < α, β > is both a ν_{α}-ideal and a ν_{β}-ideal.

(2) α and β induce the same ordering on \frac{B}{<α,β>} (in particular, the set of ν_{α}-ideals containing < α, β > coincides with the set of ν_{β}-ideals containing < α, β >).

(3) < α, β > is the smallest ideal (in the sense of inclusion), satisfying (1) and (2).

(4) If α and β have no common specialization then < α, β >= B.

Let f be a piecewise polynomial function on R^{n} and take a point α ∈ Sper A. Let the notation be as in Definition 1.1. The covering R^{n} = ⋃_{i} P_{i} induces a corresponding covering Sper A = ⋃_{i} P̃_{i}

of the real spectrum. Pick and fix an i such that α ∈ P̃_{i}. We set f_{α} := f_{i}. We refer to f_{α} as a local polynomial representative of f at α. In general, the choice of i is not uniquely determined by α. Implicit in the notation f_{α} is the fact that one such choice has been made.

In [13], Madden reduced the Pierce–Birkhoff conjecture to a purely local statement about separating ideals and the real spectrum. Namely, he showed that the Pierce-Birkhoff conjecture is equivalent to

Conjecture 2 (Pierce-Birkhoff conjecture, the abstract version) Let f be a piecewise polynomial function and α, β points in Sper A. Let f_{α} ∈ A be a local representative of f at α and f_{β} ∈ A a local representative of f at β. Then f_{α} − f_{β} ∈< α, β >.

The following slightly weaker conjecture has proved to be an extremely useful stepping stone on the way to Pierce–Birkhoff (we give the contrapositive of Madden’s original statement, since it is better adapted to our needs).

Conjecture 3 (the Separation conjecture) Let g ∈ A and let α, β ∈ Sper A be two points such that g∈/ < α, β >. Then α and β lie in the same connected component of the set Sper A\{g = 0}.

Proposition 1.5 The Separation conjecture is equivalent to the special case of the Pierce– Birkhoff conjecture in which the piecewise-polynomial function f is defined by two polynomials.

Proof: First, assume the Pierce–Birkhoff conjecture is true for any piecewise-polynomial f defined by two polynomials. Let g be as in the Separation conjecture. We argue by contradiction. Assume that α and β lie in two different connected components of Sper A \ {g = 0}. Let f be the piecewise polynomial function which is equal to g on the connected component of Sper A \ {g = 0} containing α and f = 0 elsewhere. Then by the Pierce–Birkhoff conjecture we have g − 0 = g ∈< α, β >, which gives the desired contradiction.

Conversely, assume that the Separation conjecture holds. Let f be a piecewise polynomial function defined by two distinct polynomials, f_{1} and f_{2}. Let P_{1} and P_{2} be closed semialgebraic sets such that f|_{P_{i}} = f_{i}|_{P_{i}}, i ∈ {1, 2}. Let g = f_{1} − f_{2}. Then the interior of P_{1} is disjoint from the interior of P_{2} and each is a union of connected components of Sper A \ {g = 0}. Pick a point α in the interior of P_{1} and β in the interior of P_{2}. Then α and β lie in two different connected components of Sper A \ {g = 0}, hence f_{1} − f_{2} ∈< α, β > by the separation conjecture. Thus the Pierce–Birkhoff conjecture holds for f. □

We now state

Conjecture 4 (the Connectedness conjecture) Let α, β ∈ Sper A and let g_{1}, …, g_{s} be a finite collection of elements of A\ < α, β >. Then there exists a connected set C ⊂ Sper A such that α, β ∈ C and C ∩ {g_{i} = 0} = ∅ for i ∈ {1, …, s} (in other words, α and β belong to the same connected component of the set Sper A \ {g_{1} … g_{s} = 0}).

One advantage of the Connectedness conjecture is that it is a statement about polynomials which makes no mention of piecewise polynomial functions.

The purpose of this paper is threefold. First, we prove (§2) that the Connectedness conjecture implies the Pierce-Birkhoff conjecture. This reduces the Pierce-Birkhoff conjecture to constructing, for each α, β ∈ Sper A, a connected set in Sper A having certain properties.

Secondly, we construct a class of connected subsets of Sper A which, though not in itself enough for the proof of the Pierce-Birkhoff conjecture, is the first and simplest example of the sort of connected sets we really need, and which constitutes the first step in our program for a proof of the Pierce-Birkhoff conjecture in dimension greater than 2. The precise relation of the main connectedness theorem of the present paper (Theorem 1.1) to the Pierce-Birkhoff conjecture, that is, the part of the program which is left for the subsequent papers, is explained in more detail later in this Introduction. In the forthcoming paper [12], the next paper of the series, we plan to prove the separation conjecture for n = 3 by constructing higher level connected sets in the real spectrum, based on the prototype constructed in the present paper.

Thirdly, we give two proofs of the Connectedness conjecture, one elementary and one invoking the main connectedness theorem of this paper, in the special case when one of α and β is monomial (this is explained in more detail later in this Introduction).

The rest of this paper is organized as follows. In §3 we define the valuation ν_{α} of B(α) associated to a point α of the real spectrum. The valuation ν_{α} is defined using the order ≤_{α}. It has following properties:

(1) ν_{α}(B[α]) ≥ 0

(2) If B is an R-algebra then for any positive elements y, z ∈ B(α),

ν_{α}(y) < ν_{α}(z) ⟹ y > Nz, ∀N ∈ R.  (2)

We also explain the geometric interpretation of a point of the real spectrum as a semi-curvette.

Notation: To simplify the notation, we will write x in place of (x_{1}, …, x_{n}).

For a point α of the real spectrum, let Γ_{α} denote the value group of ν_{α}. We say that α is monomial if it is given by a semi-curvette of the form t → (c_{1}t^{a_{1}}, …, c_{n}t^{a_{n}}), where c_{j} ∈ R and the a_{j} are strictly positive elements, ℚ-linearly independent elements of Γ_{α}. Note that if α is monomial then

k_{α} = R  (3)

Take a point α ∈ Sper A. For an element a ∈ Γ_{α}, let us denote the ν_{α}-ideal of value a and P_{a+} the greatest ν_{α}-ideal properly contained in P_{a}. In other words, we have

P_{a} = {y ∈ A | ν(y) ≥ a},  (4)
P_{a} = {y ∈ A | ν(y) > a}.  (5)

As an application of the above ideas, we give in §4 a proof of the Connectedness conjecture (and hence also of the Pierce–Birkhoff conjecture the abstract formulation) in the special case when for all the ν_{α}-ideals containing < α, β > we have

dim_{R} \frac{P_{a}}{P_{a+}} = 1  (6)

(as an R-vector space) and P_{a} is generated by monomials in x. In particular, this proves the Connectedness and the Pierce–Birkhoff conjectures in the case when one of α and β is a monomial point of the real spectrum.

The proof goes as follows. Let the notation be as in the statement of the Connectedness conjecture. By assumption, g_{i}∈/ < α, β > so ν_{α}(g_{i}) < ν_{α}(< α, β >). The assumption (6) and the monomiality of P_{a} for a < ν_{α}(< α, β >) implies that the R-vector space \frac{P_{a}}{P_{a+}} is generated by a natural image of a monomial in x. Hence, for 1 ≤ i ≤ s, g_{i} can be written as the sum of a dominant monomial M(i), of value ν_{α}(g_{i}) < ν_{α}(< α, β >), plus an R-linear combination of monomials of value strictly greater than ν_{α}(g_{i}). Let us denote M(i) by c_{i}x^{γ(i)}, with γ(i) ∈ ℕ^{n}, c_{i} ∈ R, and the non-dominant monomials appearing in g_{i} by c_{ji}x^{γ_{ji}}, c_{ji} ∈ R, γ_{ji} ∈ ℕ^{n}, 1 ≤ j ≤ N_{i}:

g_{i} = c_{i}x^{γ(i)} + ∑_{j=1}^{N_{i}} c_{ji}x^{γ_{ji}},

where

ν_{α} (x^{γ(i)}) < ν_{α} (x^{γ_{ji}}) , 1 ≤ j ≤ N_{i}.  (7)

Now, Proposition 1.4 (2) and the fact that ν_{α} (x^{γ(i)}) < ν_{α}(< α, β >) imply that

ν_{β} (x^{γ(i)}) < ν_{β} (x^{γ_{ji}}) , 1 ≤ j ≤ N_{i}.  (8)

By definition of < α, β >, if x_{q}∈/ < α, β > then the sign of x_{q}(α) is the same as that of x_{q}(β). Next, (2), (7) and (8) imply that

|x^{γ(i)}| ≥_{α} N |x^{γ_{ji}}| for all N ∈ R and 1 ≤ j ≤ N_{i},  (9)

and similarly for >_{β}. Let C denote the set of all the points δ ∈ Sper A such that

|c_{i}x^{γ(i)}| ≥_{δ} N_{i} |c_{ji}x^{γ_{ji}}| for 1 ≤ j ≤ N_{i},  (10)

and x_{q}(δ) has the same sign as x_{q}(α) for all q such that x_{q}∈/ < α, β >. Then α, β ∈ C and all of the g_{i} have constant sign on C, 1 ≤ i ≤ s. In §4 we give an elementary proof of the following fact:

Lemma 1.1 Fix an index l ∈ {0, …, n}. Let C denote the subset of Sper A defined by specifying sgn x_{q} (which can be either strictly positive on all of C or strictly negative on all of C) for q ∈ {1, …, l} and by imposing, in addition, finitely many monomial inequalities of the form

|d_{i}x^{λ_{i}} | ≥ |x^{θ_{i}} | , 1 ≤ i ≤ M  (11)

where d_{i} ∈ R \ {0}, λ_{i}, θ_{i} ∈ ℕ^{n} and x_{q} may appear only on the right hand side of the inequalities (38) for q ∈ {l + 1, …, n}. Then C is connected.

This completes the proof of the Connectedness conjecture in this special case. Another connected set C which can do the same job is the set of all δ such that the inequalities (7) are satisfied with α replaced by δ and x_{q}(δ) has the same sign as x_{q}(α) for all q such that x_{q}∈/ < α, β >. The connectedness of such a set C is an immediate corollary of the main connectedness theorem of the present paper — Theorem 1.1, stated below. This will be explained in more detail at the end of the paper.

In §5 we study the behaviour of certain subsets of the real spectrum under blowing up. In §6 we recall and adapt to our context some known results on resolution of singularities,

of a purely combinatorial nature. These results can be considered as a special case of the desingularization of toric varieties or Hironaka’s game [20]. Since they are easy to prove, we chose to include complete proofs. The conclusion of this section is an algorithm for resolving singularities of any binomial by iterating combinatorial (toric) blowings up along non-singular centers.

Finally, §7 is devoted to the proof of the main theorem. Let A = R[x_{1}, …, x_{n}] be a polynomial ring and let ω_{ij}, θ_{il} ∈ ℚ, i ∈ {1, …, n}, j ∈ {1, …, q}, l ∈ {1, …, u}, where q and u are positive integers.

Let ν_{δ} be the valuation associated to the point δ ∈ Sper(A), defined in §3.

Notation: We will write ν_{δ}(x) for the n-tuple (ν_{δ}(x_{1}), …, ν_{δ}(x_{n})).

Let

h_{j}(ν_{δ}(x)) = ∑_{i=1}^{n} ω_{ij}ν_{δ}(x_{i}) for j ∈ {1, …, q},

and

z_{l}(ν_{δ}(x)) = ∑_{i=1}^{n} θ_{il}ν_{δ}(x_{i}) for l ∈ {1, …, u}.

Theorem 1.1 The set

S = {δ ∈ Sper(A) | x_{i} >_{δ} 0, i ∈ {1, …, n}, h_{j}(ν_{δ}(x)) > 0, j ∈ {1, …, q},
z_{l}(ν_{δ}(x)) = 0, l ∈ {1, …, u}}  (12)

is connected in the spectral topology.

In other words, subsets of Sper A defined by finitely many ℚ-linear equations and strict inequalities on ν(x_{1}), …, ν(x_{n}) are connected.

To prove Theorem 1.1, we first reduce it to the following Proposition:

Proposition 1.6 Assume that u = 0 in (12), that is, S is defined by finitely many strict ℚlinear inequalities and no equalities. Then S is connected.

While revising this paper, we found a very simple proof of Proposition 1.6 using Lemma 1.1 (the first proof of Proposition 1.6 in the current version). However, we decided to keep our original proof because we feel that it sheds some light on the geometry of the real spectrum. Roughly speaking, this proof (now called the second proof of Proposition 1.6) can be thought of as a form of “path connectedness” of the set S. Along the way towards the second proof, we exhibit two other types of connected sets in Sper A, whose connectedness we believe to be of interest in its own right.

In the forthcoming paper [12], we will develop the theory of approximate roots of a valuation. Given a ring A and a valuation ν, non-negative on A, a family of approximate roots is a collection {Q_{i}}, finite or countable, of elements of A, having some additional properties as outlined below. A generalized monomial (with respect to a given collection {Q_{i}} of approximate roots) is, by definition, an element of A of the form ∏_{j} Q_{j}^{γ_{j}}, γ_{j} ∈ ℕ. The main defining properties of

the approximate roots are the fact that every ν-ideal I in A is generated by the generalized monomials contained in it, that is, generalized monomials ∏_{j} Q_{j}^{γ_{j}} satisfying

∑_{j} γ_{j}ν(Q_{j}) ≥ ν(I),

and the fact that for each i, Q_{i} is described by an explicit formula in terms of Q_{1}, ..., Q_{i−1}. In particular, the valuation ν is completely determined by the set {Q_{i}} and the values ν(Q_{i}).

We then show that every element g ∈ A can be written as a finite sum of the form

g = cQ^{θ} + ∑_{j=1}^{N} c_{j}Q^{δ_{j}},  (13)

where c and c_{j} are elements of A such that ν(c) = ν(c_{j}) = 0 and Q^{θ} and Q^{δ_{j}} are generalized monomials such that

ν (Q^{θ}) < ν (Q^{δ_{j}}) for 1 ≤ j ≤ N.  (14)

Now let δ ∈ Sper(A) and let ν = ν_{δ}. Then, by (2) and (14) the sign of g with respect to ≤_{δ} is determined by the sign of its leading coefficient c.

Let α and β be two points of Sper(A) having a common specialization. In [12] we will give an explicit description of the set of generalized monomials (with respect to approximate roots Q_{i} common to ν_{α} and ν_{β}) which generate the separating ideal < α, β >. Furthermore, we show that all the approximate roots Q_{i} for ν_{α} such that Q_{i}∈/ < α, β > are also approximate roots for ν_{β} and vice versa.

Now let g be as in the separation conjecture. The fact that g∈/ < α, β > implies the existence of an expression (13) in which all the approximate roots Q_{i} are common for ν_{α} and ν_{β} and the inequalities (14) hold for both ν = ν_{α} and ν = ν_{β}. The inequalities (14) can be viewed as linear inequalities on ν(Q_{1}), …, ν(Q_{t}) with integer coefficients.

To prove the separation conjecture, we look for connected sets C ⊂ Sper(A) having the following properties:

(1) α, β ∈ C;

(2) the Q_{i} appearing in (13) are approximate roots simultaneously for all the ν_{δ}, δ ∈ C;

(3) the inequalities (14) hold for ν = ν_{δ}, for all δ ∈ C;

(4) the leading coefficient c has constant sign on C.

Once such a C is found, (2) and (14) imply that the sign of g on C is constant, which proves that α and β lie in the same connected component of Sper(A) \ {g = 0}.

In [12] we will define the set

C(g, α, β) = {δ ∈ Sper(A) | Q_{i} > 0, i ∈ {1, …, t}, ∑_{i=1}^{t} ω_{ij}ν_{δ}(Q_{i}) > 0, j ∈ {1, …, q} ,

∑_{i=1}^{t} λ_{ij}ν_{δ}(Q_{i}) = 0, j ∈ {1, …, l}} ,  (15)

where ω_{ij}, λ_{ij} ∈ ℤ, all the inequalities (14) described above appear among the ∑_{i=1}^{t} ω_{ij}ν_{δ}(Q_{i}) > 0

and the remaining equalities and inequalities on the right hand side of (15) encode the fact that Q_{1}, …, Q_{t} are approximate roots for ν_{δ} for all δ belonging to C(g, α, β). If g_{1}, …, g_{s} is a finite collection of elements of A \ < α, β >, we put

C(g_{1}, …, g_{s}, α, β) = ⋂_{i=1}^{s} C(g_{i}, α, β).

By construction, both sets C(g, α, β) and C(g_{1}, …, g_{s}, α, β) contain α and β, the element g does not change sign on C(g, α, β) and none of the elements g_{1}, …, g_{s} change sign on C(g_{1}, …, g_{s}, α, β). Thus to prove the separation conjecture it is sufficient to prove that C(g, α, β) is connected and to prove the Pierce–Birkhoff conjecture it is sufficient to prove that C(g_{1}, …, g_{s}, α, β) is connected.

A proof of the connectedness of C(g, α, β) and C(g_{1}, …, g_{s}, α, β) is relegated to future papers. In any case, the most delicate part of the proof of the separation (resp. the Pierce–Birkhoff) conjecture is proving the connectedness of C(g, α, β) (resp. C(g_{1}, …, g_{s}, α, β)). The connectedness theorem of the present paper is the special case of the desired result in which the finite set {Q_{1}, …, Q_{t}} is a subset of the set of variables {x_{1}, …, x_{n}}. In [12] we plan to reduce the connectedness of C(g, α, β) to this special case using sequences of blowings up of the form described in §6: we will construct a sequence π of blowings up such that the total transform of each Q_{i}, i ∈ {1, …, t}, is a usual monomial with respect to the new coordinates, times a unit. The preimage of C(g, α, β) under π has the form (12) in the new coordinates. This will reduce the connectedness of C(g, α, β) to that of sets of the form S, proved in this paper.

We thank the CNRS and the University of Angers for supporting J. Madden’s stay in Angers during a crucial stage of our work on this paper.

We also thank the referee for a very thorough reading of the paper and for his constructive criticism which helped us improve the exposition and eliminate numerous inaccuracies.

2 The Connectedness conjecture implies the Pierce-Birkhoff conjecture

Theorem 2.1 The Connectedness conjecture implies the Pierce-Birkhoff conjecture.

Proof: We will assume the Connectedness conjecture and deduce the Pierce–Birkhoff conjecture in the form of Conjecture 2. Let f ∈ P W (A) and let {f_{i}}_{1≤i≤r} denote the elements of A which represent f on the various closed constructible subsets P_{i} ⊂ Sper A. Let α, β ∈ Sper A and let

T = {{i, j} ⊂ {1, …, r} | f_{i} − f_{j}∈/ < α, β >}.  (16)

α  β f_{i_{s−1}} f_{i_{s}} f_{i_{2}} f_{i_{1}}

We apply the Connectedness conjecture to the finite collection {f_{i} − f_{j} | {i, j} ∈ T } of elements of A. By the Connectedness conjecture, there exists a connected subset C ⊂ Sper A such that α, β ∈ C and

C ∩ {f_{i} − f_{j} = 0} = ∅ for all {i, j} ∈ T.  (17)

Let J ⊂ {1, …, r} be the set of all the indices j having the following property: there exist i_{1}, …, i_{s} ∈ {1, …, r} such that

α = i_{1},  (18)
j = i_{s}  (19)

and for each q ∈ {1, ..., s − 1}, we have

C ∩ {f_{i_{q}} − f_{i_{q+1}} = 0} ≠ ∅.  (20)

Let F = ⋃_{j∈J}(P_{j} ∩ C). We have α ∈ F by definition.

Claim: F = C; in particular, β ∈ F. Proof of Claim. Let J^{c} = {1, …, r} \ J and G = ⋃_{j∈J^{c}}(P_{j} ∩ C). Clearly

C = F ∪ G  (21)

and both sets F and G are closed in the induced topology of C (since all the P_{j} are closed). Next, we show that

F ∩ G = ∅  (22)

Indeed, if δ ∈ F ∩ G then there exist j ∈ J and j′ ∈ J^{c} such that δ ∈ P_{j} ∩ P_{j′}. But then f_{j}(δ) = f_{j′}(δ), so δ ∈ C ∩ {f_{j} − f_{j′} = 0}, hence j′ ∈ J, a contradiction. This proves (22), so the union in (21) is a disjoint union.

Now, since C is connected and F ≠ ∅ (since α ∈ F), the expression (21) of C as a disjoint union of closed sets implies that G = ∅. Hence β ∈ F, which completes the proof of the Claim.

Let j ∈ J be such that β ∈ P_{j}, so that f_{j} = f_{β}. Let i_{1}, …, i_{s} be as in (18)–(20), expressing the fact that j ∈ J. Together, (16), (17) and (20) imply that f_{i_{q}} − f_{i_{q+1}} ∈< α, β > for all q ∈ {1, ..., s − 1}. In view of (18) and (19), we obtain f_{α} − f_{β} ∈< α, β >, as desired. □

3 The valuation associated to a point in the real spectrum

Let B be a ring and α a point in Sper B. In this section we define the valuation ν_{α} of B(α), associated to α. We also give a geometric interpretation of points in Sper B as semi-curvettes.

Terminology: If B is an integral domain, the phrase “valuation of B” will mean “a valuation of the field of fractions of B, non-negative on B”. Also, we will sometimes commit the following abuse of notation. Given a ring B, a prime ideal 𝔭 ⊂ B, a valuation ν of \frac{B}{𝔭} and an element x ∈ B, we will write ν(x) instead of ν(x mod 𝔭), with the usual convention that ν(0) = ∞, which is taken to be greater than any element of the value group.

First, we define the valuation ring R_{α} by

R_{α} = {x ∈ B(α) | ∃z ∈ B[α], |x| ≤_{α} z}.

That R_{α} is, in fact, a valuation ring, follows because for any x ∈ B(α), either x ∈ R_{α} or \frac{1}{x} ∈ R_{α}. The maximal ideal of R_{α} is M_{α} = \{x ∈ B(α) | |x| < \frac{1}{|z|}, ∀z ∈ B[α] \ \{0\}\}; its residue field k_{α} comes equipped with a total ordering, induced by ≤_{α}. For a ring B let U(B) denote the

multiplicative group of units of B. Recall that Γ_{α} ≅ \frac{B(α) \ \{0\}}{U(R_{α})} and that the valuation ν_{α} can

be identified with the natural homomorphism

B(α) \ \{0\} → \frac{B(α) \ \{0\}}{U(R_{α})}.

By definition, we have a natural ring homomorphism

B → R_{α}  (23)

whose kernel is 𝔭_{α}.

Remark 3.1 ([3], [11], [5] 10.1.10, p. 217) Conversely, the point α can be reconstructed from the ring R_{α} by specifying a certain number of sign conditions (finitely many conditions when B is noetherian), as we now explain. Take a prime ideal 𝔭 ⊂ B and a valuation ν of κ(𝔭) := \frac{B_{𝔭}}{𝔭B_{𝔭}}, with value group Γ. Let

r = dim_{𝔽_{2}}(Γ/2Γ)

(if B is not noetherian, it may happen that r = ∞). Let x_{1}, …, x_{r} be elements of κ(𝔭) such that ν(x_{1}), …, ν(x_{r}) induce a basis of the 𝔽_{2}-vector space Γ/2Γ. Then for every x ∈ κ(𝔭), there exists f ∈ κ(𝔭), and a unit u of R_{ν} such that x = ux_{1}^{ϵ_{1}} ⋯ x_{r}^{ϵ_{r}}f^{2} with ϵ_{i} ∈ {0, 1} (to see this, note that for a suitable choice of f and ϵ_{j} the value of the quotient u of x by the product x_{1}^{ϵ_{1}} ⋯ x_{r}^{ϵ_{r}}f^{2} is 0, hence u is invertible in R_{ν}). Now, specifying a point α ∈ Sper B supported at 𝔭 amounts to specifying a valuation ν of \frac{B}{𝔭}, a total ordering of the residue field k_{ν} of R_{ν}, and the sign data sgn x_{1}, …, sgn x_{r}. For x∈/𝔭, the sign of x is given by the product sgn(x_{1})^{ϵ_{1}} ⋯ sgn(x_{r})^{ϵ_{r}}sgn(u), where sgn(u) is determined by the ordering of k_{ν}.

Points of Sper B admit the following geometric interpretation (we refer the reader to [7], [9], [17], p. 89 and [18] for the construction and properties of generalized power series rings and fields).

Definition 3.1 Let k be a field and Γ an ordered abelian group. The generalized formal power series ring k [[t^{Γ}]] is the ring formed by elements of the form ∑_{γ} a_{γ}t^{γ}, a_{γ} ∈ k such that the set

{γ | a_{γ} ≠ 0} is well ordered.

The ring k [[t^{Γ}]] is equipped with the natural t-adic valuation v with values in Γ, defined by v(f) = inf{γ | a_{γ} ≠ 0} for f = ∑_{γ} a_{γ}t^{γ} ∈ k [[t^{Γ}]]. Specifying a total ordering on k and dim_{𝔽_{2}}(Γ/2Γ) sign conditions defines a total ordering on k [[t^{Γ}]]. In this ordering |t| is smaller than any positive element of k. For example, if t^{γ} > 0 for all γ ∈ Γ then f > 0 if and only if a_{v(f)} > 0.

For an ordered field k, let k̄ denote the real closure of k. The following result is a variation on a theorem of Kaplansky ([9], [10]) for valued fields equipped with a total ordering.

Theorem 3.1 ([18], p. 62, Satz 21) Let K be a real valued field, with residue field k and value group Γ. There exists an injection K ↪ k̄ ((t^{Γ})) of real valued fields.

Let α ∈ Sper B and let Γ be the value group of ν_{α}. In view of (23) and the Remark above, specifying a point α ∈ Sper B is equivalent to specifying a total order of k_{α}, a morphism

B[α] → k̄_{α} [[t^{Γ}]]

and dim_{𝔽_{2}}(Γ/2Γ) sign conditions as above.

We may pass to usual spectra to obtain morphisms

Spec (k̄_{α} [[t^{Γ}]]) → Spec B[α] → Spec B.

In particular, if Γ = ℤ, we obtain a formal curve in Spec B (an analytic curve if the series are convergent). This motivates the following definition:

Definition 3.2 Let k be an ordered field. A k-curvette on Sper(B) is a morphism of the form

α : B → k [[t^{Γ}]] ,

where Γ is an ordered group. A k-semi-curvette is a k-curvette α together with a choice of the sign data sgn x_{1},..., sgn x_{r}, where x_{1}, ..., x_{r} are elements of B whose t-adic values induce an 𝔽_{2}-basis of Γ/2Γ.

We have thus explained how to associate to a point α of Sper B a k̄_{α}-semi-curvette. Conversely, given an ordered field k, a k-semi-curvette α determines a prime ideal 𝔭_{α} (the ideal of all the elements of B which vanish identically on α) and a total ordering on B/𝔭_{α} induced by the ordering of the ring k [[t^{Γ}]] of formal power series. These two operations are inverse to each other. This establishes a one-to-one correspondence between semi-curvettes and points of Sper B.

Below, we will often describe points in the real spectrum by specifying the corresponding semi-curvettes.

Example: Consider the curvette ℝ[x, y] → ℝ[[t]] defined by x ↦ t^{2}, y ↦ t^{3}, and the semicurvette given by declaring, in addition, that t is positive. This semi-curvette is nothing but the upper branch of the cusp.

Throughout this paper, we study sets of points of the real spectrum defined by certain properties of their associated valuations. As the point δ varies, so does ν_{δ} and its value group Γ_{δ}. In order to describe such sets in the real spectrum, we need to embed Γ_{δ} in some “universal” ordered group.

Notation and convention: Let us denote by Γ the ordered group ℝ_{lex}^{n}. This means that elements of Γ are compared as words in a dictionary: we say that (a_{1}, …, a_{n}) < (a′_{1}, …, a′_{n}) if and only if there exists j ∈ {1, …, n} such that a_{q} = a′_{q} for all q < j and a_{j} < a′_{j}.

The reason for introducing Γ is that by Abhyankar’s inequality we have rank ν_{δ} ≤ dim A = n for all δ ∈ Sper A, so the value group Γ_{δ} can be embedded into Γ as an ordered subgroup (of course, this embedding is far from being unique). Let Γ_{+} be the semigroup of non-negative elements of Γ.

Fix a strictly positive integer ℓ. In order to deal rigourously with ℓ-tuples of elements of Γ_{δ} despite the non-uniqueness of the embedding Γ_{δ} ⊂ Γ, we introduce the category OGM(ℓ), as follows. An object in OGM(ℓ) is an ordered abelian group G together with ℓ fixed generators a_{1}, …, a_{ℓ} (such an object will be denoted by (G, a_{1}, …, a_{ℓ})). A morphism from (G, a_{1}, …, a_{ℓ}) to (G′, a′_{1}, …, a′_{ℓ}) is a homomorphism G → G′ of ordered groups which maps a_{j} to a′_{j} for each j.

Given (G, a_{1}, …, a_{ℓ}), (G′, a′_{1}, …, a′_{ℓ}) ∈ Ob(OGM(ℓ)), the notation

(a_{1}, …, a_{ℓ}) ∼_{∘} (a′_{1}, …, a′_{ℓ})  (24)

will mean that (G, a_{1}, …, a_{ℓ}) and (G′, a′_{1}, …, a′_{ℓ}) are isomorphic in OGM(ℓ).

Take an element

a = (a_{1}, …, a_{ℓ}) ∈ Γ_{+}^{ℓ}.

Let G ⊂ Γ be the ordered group generated by a_{1}, …, a_{ℓ}. Then (G, a_{1}, …, a_{ℓ}) ∈ Ob(OGM(ℓ)). For each δ ∈ Sper(A), let Γ_{δ} denote the value group of the associated valuation ν_{δ} and Γ_{δ}^{∗} the subgroup of Γ_{δ} generated by ν_{δ}(x_{1}), …, ν_{δ}(x_{n}). In this way, we associate to δ an object (Γ_{δ}^{∗}, ν_{δ}(x_{1}), …, ν_{δ}(x_{n})) ∈ Ob(OGM(n)).

4 A proof of the Connectedness conjecture in the case when one of α and β is monomial

Consider two points α, β ∈ Sper A. For an element a ∈ Γ_{α}, let P_{a} denote the ν_{α}-ideal of value a and P_{a+} the greatest ν_{α}-ideal which is properly contained in P_{a} (cf. (4)–(5)). The purpose of this section is to prove the Connectedness (and hence also the Pierce–Birkhoff) conjecture assuming that P_{a} is generated by monomials in x whenever a ≤ ν_{α}(< α, β >) and that

dim_{R} \frac{P_{a}}{P_{a+}} = 1 whenever a < ν_{α}(< α, β >).  (25)

We start with a general (and known) result on semi-algebraic connectedness of constructible sets in the Euclidean space. We give a proof since we did not find the exact statement we needed in the literature (a slightly weaker form of it is given in [2], p. 52, Proposition–Definition 62). We state and prove the result in greater generality than needed in this section, because of the applications we have in mind later in the paper.

Consider the totally ordered set R_{∞} := {−∞}∪R∪{+∞}, equipped with the order topology. The map

ϕ(x) = −\frac{1}{1 + x} + 1 if x ≥ 0  (26)
= \frac{1}{1 − x} − 1  if x ≤ 0  (27)

defines a homeomorphism from R_{∞} to the interval [−1, 1]. Let 1 ≤ s ≤ n − 1. The homeomorphism ϕ gives rise to a homeomorphism

ϕ^{n−s} : R_{∞}^{n−s} → [−1, 1]^{n−s}

in the obvious way, that is, by applying the map ϕ separately to each component. A closed semi-algebraic subset of R^{s} × R_{∞}^{n−s} is, by definition, the preimage of a (relatively) closed semi-algebraic subset of R^{s} × [−1, 1]^{n−s} with respect to the product topology of R^{s} × [−1, 1]^{n−s}, under the map (id, ϕ^{n−s}). Let G be a semi-algebraic subset of R^{s}. A mapping

G → R_{∞}^{n−s}

is said to be semi-algebraic if its graph is a closed semi-algebraic set.

Take a semi-algebraic set G ⊂ R^{s}. Fix two continuous semi-algebraic mappings

f, g : G → R_{∞}^{n−s},

having the following property. Write f = (f_{j+1}, …, f_{n}), g = (g_{j+1}, …, g_{n}), where f_{j} and g_{j} are continuous semi-algebraic R_{∞}-valued functions on G. We require, for each (a_{1}, …, a_{s}) ∈ R^{s} and each j ∈ {s + 1, …, n}, the inequality

f_{j}(a_{1}, …, a_{s}) ≤ g_{j}(a_{1}, …, a_{s}).

Let D_{f,g} denote the semi-algebraic subset of R^{s} × R^{n−s} defined by

D_{f,g} = {(a_{1}, …, a_{n}) ∈ R^{n} | (a_{1}, …, a_{s}) ∈ G, f_{j}(a_{1}, …, a_{s}) ≤ a_{j} ≤ g_{j}(a_{1}, …, a_{s}), s < j ≤ n} .

(28)

Let p : R^{s} × R_{∞}^{n−s} → R^{s} be the canonical projection and

p_{f,g} : D_{f,g} → G  (29)

the restriction of p to D_{f,g}.

Proposition 4.1 Assume that G is semi-algebraically connected. Then so is the semi-algebraic set D_{f,g}.

Proof: Composing both f and g with the continuous semi-algebraic mapping ϕ^{n−s}, we may assume that the images of f and g lie in R^{n−s} rather than in R_{∞}^{n−s}.

The map p_{f,g} has semi-algebraically connected fibers (in fact, each fiber of p_{f,g} is the parallelepiped defined by the inequalities

f_{j}(a_{1}, …, a_{s}) ≤ x_{j} ≤ g_{j}(a_{1}, …, a_{s}), j ∈ {s + 1, …, n},

which is semi-algebraically connected ([5], Proposition 2.4.3, p. 31)). Suppose D_{f,g} were not semi-algebraically connected. Then there would exist open semi-algebraic sets U, V ⊂ R^{n} such that

D_{f,g} = (U ∩ D_{f,g}) ∐(V ∩ D_{f,g}),

with U ∩ D_{f,g} ≠ ∅, V ∩ D_{f,g} ≠ ∅. Let U_{0} = p_{f,g}(U ∩ D_{f,g}), V_{0} = p_{f,g}(V ∩ D_{f,g}).

Claim: We have U_{0} ∩ V_{0} = ∅. Proof of the Claim: We prove the Claim by contradiction. Take a point ξ ∈ U_{0} ∩ V_{0}. Then p_{f,g}^{−1}(ξ) = (p_{f,g}^{−1}(ξ) ∩ U) ∐ (p_{f,g}^{−1}(ξ) ∩ V ). We have expressed the fiber p_{f,g}^{−1}(ξ) as a disjoint union of two non-empty, relatively open semi-algebraic sets which contradicts the semi-algebraic connectedness of p_{f,g}^{−1}(ξ). The Claim is proved.

It follows from the Claim that

U ∩ D_{f,g} = p_{f,g}^{−1}(U_{0}) and  (30)
V ∩ D_{f,g} = p_{f,g}^{−1}(V_{0}).  (31)

We have

G = U_{0} ∐ V_{0}.  (32)

Moreover, (30)–(31) imply that U_{0} is the preimage of the open semi-algebraic set U under the continuous semi-algebraic mapping

(id, \frac{f + g}{2} ) : G → R^{n} = R^{s} × R^{n−s},

hence it is relatively open in G, and similarly for V_{0}. Thus (32) expresses G as a disjoint union of non-empty relatively open semi-algebraic sets. Then G is not semi-algebraically connected, which gives the desired contradiction. □

Remark 4.1 Proposition 4.1 holds just as well if we replace one or both non-strict inequalities in (28) by strict ones (the same proof applies verbatim).

Theorem 4.1 Let the notation and assumptions be as in the beginning of this section. Take a finite collection g_{1}, …, g_{s} of elements of A\ < α, β >. Then there exists a connected set C ⊂ Sper A such that α, β ∈ C and C ∩ {g_{i} = 0} = ∅ for i ∈ {1, …, s} (in other words, α and β belong to the same connected component of the set Sper A \ {g_{1} … g_{s} = 0}).

Remark 4.2 Of course, the conclusion of Theorem 4.1 remains valid if its hypotheses are satisfied after applying an R-automorphism of A.

Corollary 4.1 Assume that that at least one of α, β is monomial. Then the conclusion of the Connectedness conjecture (and hence also that of the Pierce–Birkhoff conjecture in the abstract formulation) holds for the pair α, β.

Proof of Corollary 4.1 assuming Theorem 4.1: In general, for each a ∈ Γ_{α} we have a non-canonical injection

\frac{P_{a}}{P_{a+}} ⊂ k_{α}  (33)

(of \frac{A}{P_{0+}} -vector spaces), defined by picking an element y ∈ P_{a} \ P_{a+} and for each z ∈ P_{a} sending z mod P_{a+} to the natural image of \frac{z}{y} in k_{α}. Without loss of generality, we may assume that α is monomial. Then (3), (33) and the fact that R is a subfield of \frac{A}{P_{0+}} imply (25) for all a ∈ Γ_{α}, in particular, for a < ν_{α}(< α, β >). Also, it is easy to see from the definition of monomial points of the real spectrum that the ideal P_{a} is monomial for all a ∈ Γ_{α}, in particular for a ≤ ν_{α}(< α, β >). Thus the hypotheses of Theorem 4.1 are satisfied whenever α is monomial. This proves the Corollary. □

Proof of Theorem 4.1: We have g_{i}∈/ < α, β > by assumption, so ν_{α}(g_{i}) < ν_{α}(< α, β >) (since < α, β > is a ν_{α}-ideal by Proposition 1.4 (1)). The assumption (25) and the monomiality of P_{a} for a ≤ ν_{α}(< α, β >) imply that the R-vector space \frac{P_{a}}{P_{a+}} is generated by a natural image of a monomial in x. Hence, for 1 ≤ i ≤ s, g_{i} can be written as the sum of a dominant monomial (which we will denote by c_{i}x^{γ(i)}, γ(i) ∈ ℕ^{n}, c_{i} ∈ R), of value ν_{α}(g_{i}) < ν_{α}(< α, β >), plus an R-linear combination of monomials of value strictly greater than ν_{α}(g_{i}). Let us denote these non-dominant monomials appearing in g_{i} by c_{ji}x^{γ_{ji}}, c_{ji} ∈ R, γ_{ji} ∈ ℕ^{n}, 1 ≤ j ≤ N_{i}:

g_{i} = c_{i}x^{γ(i)} + ∑_{j=1}^{N_{i}} c_{ji}x^{γ_{ji}},

where

ν_{α} (x^{γ(i)}) < ν_{α}(x^{γ_{ji}}), 1 ≤ j ≤ N_{i}.  (34)

Proposition 1.4 (2) and the fact that ν_{α} (x^{γ(i)}) < ν_{α}(< α, β >) imply that

ν_{β} (x^{γ(i)}) < ν_{β}(x^{γ_{ji}}), 1 ≤ j ≤ N_{i}.  (35)

Now, (2), (34) and (35) imply that

|x^{γ(i)}| ≥_{α} N |x^{γ_{ji}}| for all N ∈ R and 1 ≤ j ≤ N_{i},  (36)

and similarly for ≥_{β}. By definition of the separating ideal, if x_{q}∈/ < α, β > then the sign of x_{q}(α) is the same as that of x_{q}(β). Renumbering the x_{q}, if necessary, we may assume that there exists l ∈ {0, …, n} such that x_{1}, …, x_{l}∈/ < α, β > and x_{l+1}, …, x_{n} ∈< α, β >. Then for q ∈ {l + 1, …, n} the variable x_{q} appears only on the right side of the inequalities (34), (35) and (36) (otherwise we would have ν_{α}(x_{q}) < ν_{α}(< α, β >), a contradiction).

Let C denote the set of all the points δ ∈ Sper A such that

|c_{i}x^{γ(i)}| ≥_{δ} N_{i} |c_{ji}x^{γ_{ji}}| for 1 ≤ j ≤ N_{i},  (37)

x_{q}(δ) has the same sign as x_{q}(α) for all q such that x_{q}∈/ < α, β >, and for q ∈ {l + 1, …, n} the variable x_{q} appears only on the right side of the inequalities (37). Then α, β ∈ C and all of the g_{i} have constant sign on C, 1 ≤ i ≤ s. Thus to complete the proof Theorem 4.1, it remains to prove the following Lemma:

Lemma 4.1 Let C denote the subset of Sper A defined by specifying sgn x_{q} (which can be either strictly positive on all of C or strictly negative on all of C) for q ∈ {1, …, l} and by imposing, in addition, finitely many monomial inequalities of the form

|d_{i}x^{λ_{i}} | ≥ |x^{θ_{i}} | , 1 ≤ i ≤ M  (38)

where d_{i} ∈ R \ {0}, λ_{i}, θ_{i} ∈ ℚ_{+}^{n} and x_{q} may appear only on the right hand side of the inequalities (38) for q ∈ {l + 1, …, n}. Then C is connected.

Convention on rational exponents. It is convenient to allow the exponents in (38) to be rational. We ascribe the obvious meaning to such an inequality by raising both sides to a suitable integer power to make all the exponents integer.

Remark 4.3 Of course, Lemma 4.1 holds just as well if we replace the non-strict inequalities in (38) by strict ones (the proof is the same). For the proof of Theorem 4.1, it makes no difference whether we use strict or non-strict inequalities: both work equally well. We state Lemma 4.1 as we do because it will be crucial in the first proof of our main connectedness theorem — Theorem 1.1 — where non-strict inequalities are essential because there we need the set C to be closed as well as connected.

Proof of Lemma 4.1: We proceed by induction on n. If n = 1, all of the inequalities (38) amount to either |x_{1}| ≤ c or |x_{1}| ≥ d for some c, d ∈ R, with the latter type of inequality allowed only if l = 1, in which case the sign of x_{1} on C is fixed. Thus C is either empty or an interval of one of the forms {c′ ≤ x_{1} ≤ d′}, {0 < x_{1} ≤ d′} or {c′ ≤ x_{1} < 0}, c′, d′ ∈ R, which are well known to be connected.

Next, assume that n > 1 and that the result is known for n − 1. Without loss of generality, we may assume that there exists M_{1} ∈ {0, …, M} such that the inequalities (38) involve x_{n} for 1 ≤ i ≤ M_{1} and do not involve x_{n} for M_{1} < i ≤ M. First, suppose

l = n.  (39)

Let x̄ = (x_{1}, …, x_{n−1}) and for each i, 1 ≤ i ≤ M_{1}, rewrite the i-th inequality (38) in one of the forms

|x_{n}| ≤ |d′_{i}x̄λ̄_{i} | or  (40)
|x_{n}| ≥ |d′_{i}x̄λ̄_{i} |  (41)

where d′_{i} ∈ R and λ̄_{i} ∈ ℚ^{n−1}. Without loss of generality, we may assume that there exists T ∈ {0, …, M_{1}} such that for i ≤ T the i-th inequality is of the form (40) while for T < i ≤ M_{1} the i-th inequality is of the form (41). Then the inequalities (38) for 1 ≤ i ≤ M_{1} can be rewritten as

max_{T <i≤M_{1}} {|d′_{i}x̄λ̄_{i} |} ≤ |x_{n}| ≤ min_{1≤i≤T} {|d′_{i}x̄λ̄_{i} |} .  (42)

Here the first inequality in (42) is taken to be non-existent if T = M_{1} and the second inequality is considered non-existent if T = 0.

By (39), x_{n} is either strictly positive on all of C or strictly negative on all of C. Without loss of generality, we may assume that x_{n} > 0 on C. Then we may replace |x_{n}| by x_{n} in the inequalities (42). Consider the set C̃_{n−1} ⊂ Sper R[x_{1}, …, x_{n−1}], defined by the inequalities

|d′_{i}x̄λ̄_{i} | ≤ |d′_{j}x̄λ̄_{j} | , i ∈ {T + 1, …, M_{1}}, j ∈ {1, …, T },  (43)

all the inequalities (38) not involving x_{n} (that is, those with M_{1} < i ≤ M), and the same sign conditions on sgn x_{1}, …, sgn x_{n−1} as appear in the definition of C. By the induction assumption, C̃_{n−1} is a connected, constructible set of Sper R[x_{1}, …, x_{n−1}].

A subset of R^{n} defined by the conditions of the form x̄ ∈ C_{n−1},

f(x̄) ≤ x_{n} ≤ g(x̄),  (44)

where f and g are two continuous, semi-algebraic functions defined on a connected, open semialgebraic set C_{n−1} ⊂ R^{n−1} with f(x̄) ≤ g(x̄), is semi-algebraically connected by Proposition 4.1 (if T = M_{1} in (42), we take f = 0 and the first inequality in (44) becomes strict). Hence C is connected by Proposition 1.2. This proves Lemma 4.1 in the case when l = n.

If l < n, the above argument also goes through, with the following changes:

(1) Since l < n, the inequalities of the form (41) do not appear in the definition of C, so that T = M_{1}.

(2) The inequality (42) becomes

|x_{n}| ≤ min_{1≤i≤M_{1}} {|d′_{i}x̄λ̄_{i} |} .  (45)

Note that if x_{q} appears on the right hand side of (45) with a negative exponent then q < l and so x_{q} is either strictly positive or strictly negative on all of C.

(3) The sets C̃_{n−1} ⊂ Sper R[x_{1}, …, x_{n−1}] and C_{n−1} ⊂ R^{n−1} are defined by all the inequalities (38) not involving x_{n} (that is, those with M_{1} < i ≤ M), and sign conditions on sgn x_{1}, …, sgn x_{n−1} which appear in the definition of C.

(4) The functions f and g of (44) are given by

f = − min_{1≤i≤M_{1}} {|d′_{i}x̄λ̄_{i} |} and  (46)
g = min_{1≤i≤M_{1}} {|d′_{i}x̄λ̄_{i} |} ,  (47)

where if M_{1} = 0, we take f = −∞ and g = +∞. This completes the proof of Lemma 4.1 and Theorem 4.1. □

Another set C which satisfies the conclusion of Theorem 4.1 is the set of all δ ∈ Sper A such that the inequalities (34) are satisfied with α replaced by δ and x_{q}(δ) has the same sign as x_{q}(α) for all q such that x_{q}∈/ < α, β >. The connectedness of such a set C is a corollary of Theorem 1.1, as we will explain at the end of this paper.

5 Affine monomial blowings up

In this section we define one of our main technical tools: affine monomial blowing up. We describe a large class of situations in which the valuation ν_{α} is preserved under blowing up.

Notation. For a subset J ⊂ {1, …, n}, x_{J} will stand for the set {x_{q} | q ∈ J}.

Let G be an ordered group. For an n-tuple a = (a_{1}, …, a_{n}) ∈ G^{n} of elements of G, we define

rat.rk a = dim_{ℚ} ∑_{j=1}^{n} ℚa_{j} ⊂ G ⊗_{ℤ} ℚ.

Consider a set J ⊂ {1, …, n}. Fix an element j ∈ J. Let

x′_{q} = x_{q}  if q = j or q∈/J  (48)
= \frac{x_{q}}{x_{j}}  if q ∈ J \ \{j\}.  (49)

Let x = (x_{1}, …, x_{n}), x′ = (x′_{1}, …, x′_{n}) and A′ = R[x′]. We have a natural ring homomorphism π : A → A′ and the corresponding map of real spectra π^{∗} : Sper A′ → Sper A.

Remark 5.1 Since the variables x are monomials in the x′ and vice versa, we have

rat.rk(ν(x_{1}), …, ν(x_{n})) = rat.rk(ν(x′_{1}), …, ν(x′_{n})).  (50)

Definition 5.1 The map π is called an affine monomial blowing up of Sper A (along the ideal (x_{J})). The choice of the set J, or the ideal (x_{J}), is referred to as the choice of the center of blowing-up and the choice of j ∈ J — as the choice of a coordinate chart. Finally, let 𝔭 be a prime ideal of A, not containing any of x_{1}, …, x_{n}. Let ν be a valuation of \frac{A}{𝔭}. We say that π is an affine monomial blowing up with respect to ν if

ν(x_{j}) = min{ν(x_{q}) | q ∈ J}.  (51)

Consider an index q ∈ {1, …, n}. We say that π is independent of x_{q} if q∈/J.

Remark 5.2 (1) Let 𝔭′ denote the strict transform of 𝔭 in Spec A′. Condition (51) is equivalent to saying that

∀g′ ∈ \frac{A′}{𝔭′} we have ν_{δ}(g′) ≥ 0.  (52)

(2) In the notation of Definition 5.1, suppose that π is independent of x_{q}. Then x_{q} = x′_{q}. In this way, we can define the notion of a sequence of affine blowings up, independent of x_{q}, recursively in the length of the sequence. Namely, if π : A → A′ is a sequence of affine blowings up of length t, independent of x_{q}, x_{q} equals the q-th coordinate of A′ and π′ : A′ → A′′ is an affine blowing up independent of x_{q} then we say that π̃ = π′ ∘π is a sequence of affine blowings up, independent of x_{q}.

We consider the coordinates x′ as part of the data of the affine monomial blowing up π. An affine monomial blowing up is completely determined by the choice of J and j as above.

We will use the following notation. For a set E ⊂ Γ_{+}^{n}, let

S_{E} = {δ ∈ Sper(A) | x_{i} >_{δ} 0, i ∈ {1, …, n}, ∃a = (a_{1}, …, a_{n}) ∈ E such that

(ν_{δ}(x_{1}), …, ν_{δ}(x_{n})) ∼_{∘} (a_{1}, …, a_{n})}.

and in particular, for a ∈ Γ_{+}^{n} we will write

S_{a} = {δ ∈ Sper(A) | x_{i} >_{δ} 0, i ∈ {1, …, n}, (ν_{δ}(x_{1}), …, ν_{δ}(x_{n})) ∼_{∘} (a_{1}, …, a_{n})}

We will need the following comparison result which says that blowing up induces a homeomorphism on sets of the form S_{E}.

Let E be a subset of Γ_{+}^{n}. Take a subset J ⊂ {1, …, n}. Pick a j ∈ J and consider the affine monomial blowing up

π : A → A′ = R[x′],

determined by J and j. Assume that π is an affine monomial blowing up with respect to all ν_{δ}, δ ∈ S_{E} (in other words, ν_{δ}(x_{j}) = min{ν_{δ}(x_{q})}_{q∈J} for all δ ∈ S_{E}; we have ν_{δ}(x′_{i}) ≥ 0 for all δ ∈ S_{E} and i ∈ {1, …, n}).

For a ∈ Γ_{+}^{n}, let a′ be the element of Γ_{+}^{n} defined by

a′_{q} = a_{q}  if q∈/J or q = j  (53)
a′_{q} = a_{q} − a_{j} if q ∈ J \ {j}.  (54)

Let E′ = {a′ | a ∈ E} and let S_{E′} denote the corresponding subset of Sper A′. Let

π^{∗} : Sper A′ → Sper A

be the map of real spectra induced by π. It is well known and easy to see that π^{∗} is a homeomorphism away from the zero set V (x_{j}) of the ideal (x_{j}). Since S_{E} is disjoint from V (x_{j}), π^{∗} induces a homeomorphism π^{∗}|_{(π^{∗})^{−1}(S_{E})} : (π^{∗})^{−1}(S_{E}) →^{∼} S_{E}. For δ ∈ Sper A \ V (x_{j}), let δ′ denote the unique preimage of δ in Sper A′.

Proposition 5.1 Take a ∈ Γ_{+}^{n} and δ ∈ Sper A \ V (x_{j}). Assume that π is an affine blowing up with respect to ν_{δ}. Then

ν_{δ} = ν_{δ′}  (55)

and

δ′ ∈ S_{a′} ⟺ δ ∈ S_{a}.  (56)

Proof: To prove (55), it suffices to show that R_{δ} = R_{δ′}. By definition,

R_{δ} = \{f ∈ κ(𝔭_{δ}) | ∃g ∈ \frac{A}{𝔭_{δ}} , |f| ≤ g \}  (57)

R_{δ′} = \{f ∈ κ(𝔭_{δ}) | ∃g′ ∈ \frac{A′}{𝔭_{δ′}} , |f| ≤ g′ \} .  (58)

Since \frac{A}{𝔭_{δ}} ↪ \frac{A′}{𝔭_{δ′}}, we have R_{δ} ⊂ R_{δ′}. To prove the opposite inclusion, note that, since π is an affine blowing up with respect to ν_{δ}, we have \frac{A′}{𝔭_{δ′}} ⊂ R_{δ} by (52). Hence, ∀g′ ∈ \frac{A′}{𝔭_{δ′}} , ∃g ∈ \frac{A}{𝔭_{δ}} such that |g′| ≤ g. This proves that R_{δ′} ⊂ R_{δ} as desired.

(56) follows from the equations (48)–(49) and (53)–(55). □

Corollary 5.1 Assume that π is an affine blowing up with respect to ν_{δ} for all δ ∈ S_{E}. We have π^{∗}(S_{E′}) = S_{E} and the restriction of π^{∗} to S_{E′} is a homeomorphism.

Definition 5.2 The point δ′ is called the transform of δ. Similarly, a′, E′ and S_{E′} are called transforms of a, E and S_{E}, respectively.

For future reference, we will also define the transform of a ℚ-linear relation on ν_{δ}(x_{1}), …, ν_{δ}(x_{n}). Consider a ℚ-linear equality of the form

∑_{i=1}^{n} θ_{i}a_{i} = 0.  (59)

Definition 5.3 The transform of (59) under π is the equality ∑_{i=1}^{n} θ′_{i}a′_{i} = 0, where

θ′_{i} = θ_{i}  if i ≠ j  (60)
= ∑_{q∈J} θ_{q} if i = j.  (61)

The transforms of all the above objects under sequences of blowings up are defined in the obvious way (that is, as iterated transforms) recursively with respect to the length of the blowing up sequence.

Example: This example shows that Proposition 5.1 (resp. Corollary 5.1) is false without the assumption that π is an affine monomial blowing up with respect to ν_{δ} (resp. with respect to ν_{δ} for all δ ∈ S_{E}). Let n = 2 and a = ((1, 0), (1, 0)) ∈ Γ^{2}. Consider the point α ∈ Sper A given by the semi-curvette α = (t^{(1,0)}, t^{(0,1)}). Let J = \{1, 2\}, j = 1, x′_{1} = x_{1}, x′_{2} = \frac{x_{2}}{x_{1}} . Since (1, 0) > (0, 1) (see the definition of lexicographical in §3), the corresponding blowing up π : A → A′ is not an affine blowing up with respect to ν_{α}. We have 𝔭_{α′} = (0), R_{α′} = R(x′_{2})[x_{1}]_{(x_{1})}, x_{1} > 0, and the total ordering on the residue field R(x′_{2}) is given by the inequalities c < x′_{2} for all c ∈ R (this information describes α′ completely). We have 0 < x′_{1} < c < x′_{2} for any positive constant c ∈ R. Moreover, ν_{α′}(x′_{2}) = (0, 0), ν_{α′}(x′_{1}) = (1, 0), so ν_{α′} ≠ ν_{α}. We have α′ ∈ S_{a′} but α∈/S_{a}, so the analogues of (56) and Corollary 5.1 do not hold without the assumption that π is an affine monomial blowing up with respect to ν_{δ}.

6 Desingularization of binomials by monomial blowings up

In this section, we recall and adapt to our context a result from the theory of resolution of singularities—a version of local uniformization of toric varieties by blowings up along smooth centers, or Hironaka’s game.

Proposition 6.1 Consider two n-tuples α, γ ∈ ℕ^{n} and the corresponding monomials x^{α} and x^{γ}. Then there exist finitely many sequences π_{i} : A → A′_{i}, 1 ≤ i ≤ s, of affine monomial blowings up, having the following properties:

(1) For each i ∈ {1, …, s}, one of x^{α} and x^{γ} divides the other in A′_{i}.

(2) For any prime ideal 𝔭 of A not containing x_{1}, …, x_{n}, and any valuation ν of \frac{A}{𝔭}, there exists i ∈ {1, …, s} such that π_{i} is a sequence of affine monomial blowings up with respect to ν.

(3) Assume that there is an index q such that α_{q} = γ_{q} = 0. Then all the blowing up sequences π_{i} above can be chosen to be independent of x_{q}.

Remark 6.1 (1) Proposition 6.1 is a special case of [20]. We give a proof here since it is much simpler than that of [20].

(2) Let ν be a valuation as in Proposition 6.1, and i ∈ {1, …, s} an index satisfying the conclusion (2) of the Proposition for ν. Then ν is non-negative on A′_{i}. If ν(x^{α}) < ν (x^{γ}), we know which of x^{α}, x^{γ} divides the other in A′_{i}, namely, x^{α} | x^{γ} (and not the other way around).

Proof of Proposition 6.1: (1) and (2) We will define a numerical character τ(α, γ), consisting of a pair of non-negative integers, associated to the unordered pair (α, γ). If one of x^{α}, x^{γ} divides the other, there is nothing to prove. Assume that neither of x^{α}, x^{γ} divides the other. We will describe a subset J ⊂ {1, …, n}, such that for any choice of j ∈ J and the corresponding affine monomial blowing up A → A′, writing x^{α} = x′^{α′}, x^{γ} = x′^{γ′} in the new coordinates, we have

τ (α′, γ′) < τ(α, γ)  (62)

in the lexicographical ordering. Since for any prime ideal 𝔭 of A not containing x_{1}, …, x_{n}, and any valuation ν of \frac{A}{𝔭}, there exists j ∈ J such that the corresponding affine monomial blowing up A → A′ is a blowing up with respect to ν (see (51)), Proposition 6.1 (1) and (2) will follow immediately by iterating this procedure.

We start by defining the numerical character τ(α, γ). Let

α = (α_{1}, …, α_{n}) and  (63)
γ = (γ_{1}, …, γ_{n}).  (64)

Let δ_{q} = min{α_{q}, γ_{q}}, 1 ≤ q ≤ n; let δ := (δ_{1}, …, δ_{n}). Let α̃ := α − δ, γ̃ = γ − δ.

Remark 6.2 With a view to proving (3) of the Proposition, note that if α_{q} = γ_{q} = 0, we have

δ_{q} = α̃_{q} = γ̃_{q} = 0.  (65)

Interchanging α and γ, if necessary, we may assume that |α̃| ≤ |γ̃| (here and below, | | stands for the sum of the components). Put

τ(α, γ) := (|α̃|, |γ̃|) .

Since we are assuming that x^{α} |̸ x^{γ} in A, we have α̃ ≠ (0, …, 0) (equivalently, |α̃| > 0). We will now describe a subset J ⊂ {1, …, n}, such that for any choice of j ∈ J and the corresponding affine monomial blowing up π : A → A′ along (x_{J}), the inequality (62) holds.

Write α̃ = (α̃_{1}, …, α̃_{n}), γ̃ = (γ̃_{1}, …, γ̃_{n}). Renumbering the variables, we may assume that there exists a, 1 ≤ a < n, such that α̃_{j} = 0 for a < j ≤ n and γ̃_{j} = 0 for 1 ≤ j ≤ a. In other words,

α̃ = (α̃_{1}, …, α̃_{a}, \underbrace{0, …, 0}_{n−a zeroes})  (66)
γ̃ = (\underbrace{0, …, 0}_{a zeroes}, γ̃_{a+1}, …, γ̃_{n}).  (67)

We may also assume that

α̃_{i} > 0 for 1 ≤ i ≤ a.  (68)

Let J denote a minimal subset of {1, …, n} (in the sense of inclusion), having the following properties:

{1, …, a} ⊂ J and  (69)
∑_{q∈J} γ̃_{q} ≥ |α̃|  (70)

Pick a j ∈ J. Let π : A → A′ be the affine monomial blowing up along (x_{J}), associated to j and J.

Remark 6.3 If α_{q} = γ_{q} = 0, (68) implies that q > a. Then by the minimality of J and (65), π is independent of x_{q}.

We will now write out the monomials x^{α} and x^{γ} in the new coordinates and observe that the numerical character τ has strictly decreased. Define the non-negative integers α̃′_{q} and γ̃′_{q}, 1 ≤ q ≤ n, as follows:

α̃′_{q} = α̃_{q}  if q ≠ j  (71)
= 0  if q = j  (72)
γ̃′_{q} = γ̃_{q}  if q ≠ j  (73)
= ∑_{q∈J} γ̃_{q} − |α̃|  if q = j.  (74)

Put α̃′ = (α̃′_{1}, …, α̃′_{n}), γ̃′ = (γ̃′_{1}, …, γ̃′_{n}). Let δ′ denote the n-vector obtained from δ by adding |α̃| to the j-th component; that is,

δ′ = (δ_{1}, …, δ_{j−1}, δ_{j} + |α̃|, δ_{j+1}, …, δ_{n}).

With these definitions, we have

x^{α} = (x′)^{δ′+α̃′}  (75)
x^{γ} = (x′)^{δ′+γ̃′}.  (76)

Put α′ = δ′ + α̃′, γ′ = δ′ + γ̃′.

Lemma 6.1 We have τ(α′, γ′) < τ(α, γ) in the lexicographical ordering.

Proof: There are two possibilities: either j ∈ {1, …, a} or j ∈ {a + 1, …, n}. If j ∈ {1, …, a} then (68), (71) and (72) imply that

|α̃′| = |α̃| − α̃_{j} < |α̃|.  (77)

Suppose that j ∈ {a + 1, …, n}. Then by (71),

|α̃′| = |α̃|.  (78)

We will prove that |γ̃′| < |γ̃|. Indeed, by the minimality of J,

∑_{q∈J\{j}} γ̃_{q} < |α̃|  (79)

(otherwise we could replace J by J \ {j}). Now, by (73), (74) and (79),

|γ̃′| = n ; ∑ ; \begin{matrix}q = a + 1 \\ q ≠ j \end{matrix} γ̃′_{q} + γ̃′_{j} = n ; ∑ ; \begin{matrix}q = a + 1 \\ q ≠ j \end{matrix} γ̃_{q} + ∑_{q∈J} γ̃_{q} − |α̃|

= ∑_{q=a+1}^{n} γ̃_{q} + ( ∑_{q∈J\{j}} γ̃_{q} − |α̃|) < ∑_{q=a+1}^{n} γ̃_{q} = |γ̃|.  (80)

To summarize, (77), (78) and (80) say that

(|α̃′|, |γ̃′|) < (|α̃|, |γ̃|) = τ(α, γ)  (81)

in the lexicographical ordering. If |α̃′| ≤ |γ̃′| then according to our definitions

τ(α′, γ′) = (|α̃′|, |γ̃′|),

and the Lemma follows from (81). If |γ̃′| < |α̃′| then τ(α′, γ′) = (|γ̃′|, |α̃′|) < (|α̃′|, |γ̃′|), and, again, the Lemma follows from (81). This completes the proof of Lemma 6.1. □

For as long as neither of x^{α}, x^{γ} divides the other, we can iterate the construction of Lemma 6.1. Since τ cannot decrease indefinitely, this process must stop after finitely many steps. Therefore after a finite number of steps we will arrive at the situation when one of x′^{α′}, x′^{γ′} divides the other. In other words, one of x^{α}, x^{γ} divides the other in A′.

Of course, the above construction is not unique: at each step we made an arbitrary choice of a coordinate chart. The inequality (62) and hence the final conclusion that one of x^{α}, x^{γ} divides the other hold for all the possible choices of j. Now let {π_{i} : A → A′_{i}}_{1≤i≤s} be the totality of all the blowing up sequences constructed above for all the possible choices of coordinate charts, such that one of x^{α}, x^{γ} divides the other in A′_{i} for all i ∈ {1, …, s} (since the number of choices of coordinate charts is finite at each step and each sequence stops after finitely many steps, the overall set is finite). Since for each valuation ν there always exists a choice of coordinate chart satisfying (51) at each step, the set {π_{i} : A → A′_{i}}_{1≤i≤s} satisfies the conclusion of the Proposition. (3) of the Proposition holds by construction according to Remarks 6.2 and 6.3. This completes the proof of Proposition 6.1. □

Remark 6.4 Keep the above notation. Take an index i ∈ {1, …, s} and consider the blowing up sequence

π_{i} : A → A′_{i},  (82)

as above. Let π : B = R[y_{1}, …, y_{n}] → B′ = R[y′_{1}, …, y′_{n}] be one of the affine blowings up appearing in the blowing up sequence (82). Let J be the subset of {1, …, n} such that π is the blowing up along the ideal (y_{J}). The definition of J in the proof of Proposition 6.1 depends only on the monomials x^{α} and x^{γ}, written in the y coordinates, and makes no mention of any valuation whatsoever. Now, take a prime ideal 𝔭 of A not containing x_{1}, …, x_{n}, and let ν be a valuation of \frac{A}{𝔭}. Let 𝔭_{B} be the strict transform of 𝔭 in B. Since, by construction, y_{1}, …, y_{n} are monomials in x_{1}, …, x_{n} (with integer exponents), we have y_{1}, …, y_{n}∈/𝔭_{B}, so ν is also a valuation of \frac{B}{𝔭_{B}}. Once the set J is defined, the choice of the coordinate chart B′ in such a way that π is an affine blowing up with respect to ν depends on ν. Proposition 6.1 (2) asserts that such a choice of coordinate chart is always possible at every step of the definition of the sequence π_{i}. Namely, we can always choose an index j ∈ J satisfying ν(y_{j}) = min_{q∈J}{ν(y_{q})}.

Fix a positive integer u ≤ n and rational numbers θ_{jl}, l ∈ {1, …, u}, j ∈ {1, …, n}, such that the matrix (θ_{jl}) has rank u. Let r := n − u. Let S be a subset of Sper A, such that for all the δ ∈ S we have x_{j} >_{δ} 0, j ∈ {1, …, n}, and

∑_{j=1}^{n} θ_{jl}ν_{δ}(x_{j}) = 0,  l ∈ {1, …, u}.  (83)

Corollary 6.1 There exist finitely many sequences π_{i} : A → A′_{i} = R[x_{i1}, …, x_{in}], 1 ≤ i ≤ s, of affine monomial blowings up having the following property. For each δ ∈ S there exists i ∈ {1, …, s} such that the following conditions hold:

(1)

ν_{δ} (x_{i,r+1}) = ⋯ = ν_{δ} (x_{in}) = 0.  (84)

(2) π_{i} is a sequence of affine monomial blowings up with respect to ν_{δ}.

Suppose there is an index q ∈ {1, …, n} such that θ_{ql} = 0 for all l ∈ {1, …, u}. Then all the affine blowing up sequences π_{i} can be chosen to be independent of x_{q}.

Proof: We proceed by induction on u. If u = 0, we have r = n and there is nothing to prove. Assume that u > 0, so r < n. Consider the non-trivial ℚ-linear relation

∑_{j=1}^{n} θ_{j1}ν_{δ}(x_{j}) = 0,  (85)

satisfied by all the points δ ∈ S. For each j ∈ {1, …, n} let

α_{j} = max{θ_{j1}, 0}  (86)
γ_{j} = max{−θ_{j1}, 0}.  (87)

Put α = (α_{1}, …, α_{n}); γ = (γ_{1}, …, γ_{n}). The equation (85) says that ν_{δ} (x^{α}) = ν_{δ}(x^{γ}) for all δ ∈ S. Apply Proposition 6.1 to the pair of monomials x^{α}, x^{γ}. Take a point δ ∈ S and let A → A′ = R[x′_{1}, …, x′_{n}] be a sequence of affine monomial blowings up with respect to ν_{δ}, appearing in the conclusion of Proposition 6.1 for x^{α} and x^{γ}. We have either \frac{x^{α}}{x^{γ}} ∈ A′ or \frac{x^{γ}}{x^{α}} ∈ A′. The fact that π is a sequence of blowings up with respect to ν_{δ} means that

ν_{δ}(x′_{j}) ≥ 0  for 1 ≤ j ≤ n.  (88)

Since \frac{x^{α}}{x^{γ}} is a monomial in x′_{1}, …, x′_{n} and since

ν_{δ} (\frac{x^{α}}{x^{γ}} ) = ν_{δ} (\frac{x^{γ}}{x^{α}} ) = 0,

(88) implies that one of the ν_{δ} (x′_{q}) is equal to zero. Renumbering the x′_{q}, if necessary, we may assume that ν_{δ}(x′_{n}) = 0. Moreover, ν_{δ} (x′_{1}) , …, ν_{δ} (x′_{n−1}) satisfy u − 1 linearly independent ℚ-linear relations (namely, the transforms of the relations (83)). Now let

π_{i} : A → A′_{i} = R[x_{i1}, …, x_{in}], 1 ≤ i ≤ s,

be the totality of affine monomial blowing up sequences satisfying the conclusion of Proposition 6.1, applied to the pair of monomials x^{α}, x^{γ} above. For i, 1 ≤ i ≤ s, let S_{i} denote the set of all the points δ ∈ S such that π_{i} is a sequence of affine monomial blowings up with respect to δ.

By Proposition 6.1 (2), we have S = ⋃_{i=1}^{s} S_{i}. Let S′_{i} denote the transform of S_{i} in Sper A′_{i}.

At this point, we have constructed finitely many affine monomial blowing up sequences π_{i}, 1 ≤ i ≤ s, having the following properties:

(1) For each δ ∈ S there exists i ∈ {1, …, s} such that δ ∈ S_{i}.

(2) For each i ∈ {1, …, s} and δ ∈ S_{i} we have

ν_{δ}(x_{in}) = 0  (89)

and ν_{δ} (x_{i1}) , …, ν_{δ} (x_{i,n−1}) satisfy u − 1 linearly independent ℚ-linear relations

∑_{j=1}^{n−1} θ_{jl}^{(i)}ν_{δ}(x_{in}) = 0  (90)

(namely, the transforms of the relations (83), from which we may delete ν_{δ}(x_{in}) in view of (89)).

Next, for each i ∈ {1, …, s}, apply the induction hypothesis to A′_{i}. Since ν_{δ}(x_{in}) does not appear in the relations (90), all the subsequent blowings up are independent of x_{q}. In particular, the equality (89) is preserved by all the subsequent affine monomial blowings up with respect to δ. This completes the proof of the Corollary. □

Remark 6.5 Let the notation be as in Corollary 6.1. Take an index i ∈ {1, …, s} and consider the blowing up sequence

π_{i} : A → A′_{i},  (91)

as in the Corollary. Let B = R[y_{1}, …, y_{n}] be a polynomial ring appearing at some stage of the blowing up sequence (91). This means that π_{i} can be written as a composition of affine monomial blowing up sequences of the form A →^{λ} B →^{λ′} A′_{i}. Let S(B) denote the subset of S consisting of all the points δ ∈ S such that λ is a sequence of affine monomial blowings up with respect to ν_{δ}. The transforms of the linear relations (83) in B are well defined for all δ ∈ S(B) and are independent of the choice of δ ∈ S(B). When we are at stage B of the construction of Corollary 6.1, the set J of the next blowing up is defined in a way independent of the choice of δ ∈ S(B); the definition of J depends only on the transforms of the linear relations (83) (Remark 6.4 and the proof of Corollary 6.1). For each j ∈ J, let μ_{j} : B → B_{j} be the blowing up determined by the choice of the coordinate chart corresponding to j. For δ ∈ S(B), we have δ ∈ S(B_{j}) if and only if μ_{j} is an affine blowing up with respect to ν_{δ}. Since for each δ ∈ S(B) we can always choose an index j ∈ J satisfying ν(y_{j}) = min_{q∈J}{ν(y_{q})}, we have S(B) = ⋃_{j∈J} S(B_{j}).

Corollary 6.2 Let the notation be as in Corollary 6.1. Take δ ∈ S such that

r = rat.rk(ν_{δ}(x_{1}), …, ν_{δ}(x_{n}))  (92)

and an index i ∈ {1, …, s} such that π_{i} is a sequence of affine monomial blowings up with respect to δ. Then ν_{δ}(x_{i1}), …, ν_{δ}(x_{ir}) are ℚ-linearly independent.

Proof: This follows immediately from (50) and (92).

7 Connectedness theorems for sets in Sper A.

As usual, let us denote by Γ the ordered group ℝ_{lex}^{n} and by Γ_{+} the semigroup of non-negative elements of Γ. Let ω_{ij}, θ_{il} ∈ ℚ, i ∈ {1, …, n}, j ∈ {1, …, q}, l ∈ {1, …, u} and consider the subset E of Γ_{+}^{n} defined by

E = {(a_{1}, …, a_{n}) ∈ Γ_{+}^{n} | ∑_{i=1}^{n} ω_{ij}a_{i} > 0, j ∈ {1, …, q}, ∑_{i=1}^{n} θ_{il}a_{i} = 0, l ∈ {1, …, u}} . (93)

Theorem 7.1 Let A = R[x_{1}, …, x_{n}] be a polynomial ring. Then the set S_{E} is connected for the spectral topology of Sper(A).

A proof of Theorem 7.1 will be given after a few lemmas.

Lemma 7.1 Let X be a topological space, normal and compact (not necessarily Hausdorff) and let F be a filter of non-empty closed connected sets. Then the intersection C = ⋂_{F ∈F} F is

non-empty, closed and connected.

Proof: That C is non-empty and closed is well known and easy to see. To prove connectedness, suppose C = X_{1} ∐ X_{2}, X_{1}, X_{2} closed and non-empty. By normality, there are two open sets U_{1} ⊃ X_{1}, U_{2} ⊃ X_{2}, U_{1} ∩ U_{2} = ∅. For any F ∈ F, Let G(F) = F \ (U_{1} ∪ U_{2}). We have G(F) ≠ ∅ because F is connected. By the compactness of X, we have ⋂_{F ∈F} G(F) ≠ ∅. Take an element

x ∈ ⋂_{F ∈F} G(F). Then

x ∈ ⋂_{F ∈F} G(F) ⊂ ⋂_{F ∈F} F = X_{1} ∐ X_{2},

but x∈/U_{1} ∪ U_{2} ⊃ X_{1} ∐ X_{2}, which is a contradiction. □

Lemma 7.2 Let B be a finitely generated R-algebra and S a closed connected subset of Sper(B). Then S is the intersection of a filter of closed connected constructible sets.

Proof: Consider the filter F of all the closed connected constructible sets containing S. The filter F is not empty because it contains the connected component of Sper B containing S; such a connected component is constructible by Proposition 1.3 and Remark 1.2.

We want to show that S = ⋂_{F ∈F} F. Suppose that S ⫋ ⋂_{F ∈F} F and take a point α ∈ ⋂_{F ∈F} F \ S.

There exists a basic open set U ∋ α such that U ∩ S = ∅. Then the connected component C of the complement of U, containing S, is a constructible closed connected set containing S, hence a member of F. But α∈/C, contradicting α ∈ ⋂_{F ∈F} F. □

In the next lemma, we will use the following notation. Let

G = R^{s} \ {∏_{j=1}^{s} x_{j} = 0} ≅ Maxr R[x_{1}, …, x_{s}]_{x_{1}···x_{s}}.

Let f, g : G → R_{∞}^{n−s} be two continuous semi-algebraic functions. Let D_{f,g} and p_{f,g} be as in (28)–(29). Let D̃_{f,g} be the constructible subset of Sper A corresponding to D_{f,g} under the bijection between semi-algebraic subsets of R^{n} and constructible subsets of Sper A, described in the Introduction. Let p̃_{f,g} : D̃_{f,g} → G̃ be corresponding map of sets in the real spectrum.

Lemma 7.3 (the Projection lemma)

If C_{0} is a closed connected subset of Sper R[x_{1}, …, x_{s}]_{x_{1}...x_{s}}, then the set

C_{f,g} := p̃_{f,g}^{−1}(C_{0})  (94)

is connected in Sper R[x_{1}, …, x_{n}]_{x_{1}...x_{s}}.

Proof of Lemma 7.3: According to the previous Lemma, we can write C_{0} as the intersection of a filter of closed connected constructible sets F̃ ∈ F. Take an F̃ ∈ F. Since F̃ is connected in the real spectrum, its preimage F in the maximal spectrum R^{s} \ {∏_{j=1}^{s} x_{j} = 0} is semi-algebraically connected ([5], Proposition 7.5.1, p. 130).

By Proposition 4.1 the closed semi-algebraic subset p_{f,g}^{−1}(F) ⊂ D_{f,g} is semi-algebraically connected.

Since p_{f,g}^{−1}(F) is semi-algebraic and semi-algebraically connected, p̃_{f,g}^{−1}(F̃) is also connected for the spectral topology (applying Proposition 7.5.1 of [5] once again). Since

C_{f,g} = ⋂_{F̃∈F} p̃_{f,g}^{−1}(F̃),

C_{f,g} is connected by Lemma 7.1.

□

Remark 7.1 By Remark 4.1 the Projection Lemma remains true if one or both strict inequalities in the definition of D̃_{f,g} and C_{f,g} (28) are replaced by non-strict ones, since the above proof applies verbatim also in that case.

Let the notation be as in Lemma 7.3.

Lemma 7.4 Let C = {δ ∈ p̃^{−1}(C_{0}) | x_{q} > 0, ν_{δ}(x_{q}) = 0, s < q ≤ n}. Then C is connected.

Proof of Lemma 7.4: For each δ ∈ C, we have

ν_{δ}(x_{s+1}) = ⋯ = ν_{δ}(x_{n}) = 0  (95)

by definition of C. (95) is equivalent to saying that the images of \frac{1}{x_{s+1}} , …, \frac{1}{x_{n}} in A(δ) lie in R_{δ}, that is, that there exists a polynomial z ∈ A such that \frac{1}{x_{q}} ≤_{δ} z, s < q ≤ n. Take j ∈ \{1, …, n\} such that x_{j} ≥_{δ} x_{q}, q ∈ {1, …, n}. Without loss of generality, we may take z to be of the form z = Nx_{j}^{b}, where N ∈ R and b ∈ ℕ. For N ∈ R with N ≥ 2, b ∈ ℕ and j ∈ {1, …, n}, let S_{N,b,j} denote the set

S_{N,b,j} = \{δ ∈ C | \frac{1}{x_{q}} ≤_{δ} Nx_{j}^{b}, q ∈ \{s + 1, …, n\}\} .

Thus for each δ ∈ C there exist N ∈ R with N ≥ 2, b ∈ ℕ and j ∈ {1, …, n} such that δ ∈ S_{N,b,j}. In other words, C = ⋃_{N,b,j} S_{N,b,j}. Furthermore, take a point δ_{0} ∈ C_{0} and let

(x_{1}(t), …, x_{s}(t)) denote the curvette representing δ_{0}. Then the curvette (x_{1}(t), …, x_{n}(t)) with x_{q}(t) = 1, s < q ≤ n, lies in S_{N,b,j} for all the choices of N, b, j as above. Thus to prove the connectedness of C, it is sufficient to prove the connectedness of S_{N,b,j} for all N, b, j.

Fix N, b and j as above. We will now use the Projection Lemma to prove that S_{N,b,j} is connected. First, suppose j ∈ {1, …, s}. Apply the Projection Lemma with

f(x_{1}, …, x_{s}) = (\frac{1}{Nx_{j}^{b}} , …, \frac{1}{Nx_{j}^{b}} )

and g = (+∞, …, +∞). Let D_{f,g}^{•} denote the subset of D_{f,g} defined by replacing the second non-strict inequality in (28) by the strict one. We have

S_{N,b,j} = (p̃_{f,g}^{•})^{−1}(C_{0}),

where p̃_{f,g}^{•} = p̃|D̃_{f,g}^{•}. Then S_{N,b,j} is connected by the Projection Lemma and Remark 7.1. Next, assume that j ∈ {s + 1, …, n}. First, let

p̃_{j} : Sper R[x_{1}, …, x_{s}, x_{j}]_{x_{1}...x_{s}} → Sper R[x_{1}, …, x_{s}]_{x_{1}...x_{s}}

be the natural projection. Let

C_{N,b,j} = \{δ ∈ p̃_{j}^{−1}(C_{0}) | x_{j}^{b+1} ≥ \frac{1}{N} \} .

Applying the Projection Lemma and Remark 7.1 to p̃_{f,g}^{•} with f = N^{− \frac{1}{b+1}} , g = +∞, we see that the set C_{N,b,j} is connected. Moreover, C_{N,b,j} is closed since it is the intersection of the preimage of the closed set C_{0} under the continuous map p̃_{j} and the closed set given by the inequality x_{j}^{b+1} ≥ \frac{1}{N}.

Next, consider the natural projection

p̃: Sper R[x_{1}, …, x_{n}]_{x_{1}...x_{s}x_{j}} ⟶ Sper R[x_{1}, …, x_{s}, x_{j}]_{x_{1}...x_{s}x_{j}}

We apply the Projection Lemma once again with s replaced by s + 1,

f(x_{1}, …, x_{s}, x_{j}) = (\frac{1}{Nx_{j}^{b}} , …, \frac{1}{Nx_{j}^{b}} )

and g = (+∞, …, +∞). We have

S_{N,b,j} = (p̃_{f,g}^{•})^{−1}(C_{N,b,j}).

Then S_{N,b,j} is connected by the Projection Lemma and Remark 7.1. This completes the proof of Lemma 7.4. □

We need one more general lemma about closures of subsets of Sper A given by finitely many strict ℚ-linear inequalities on ν(x_{1}), …, ν(x_{n}).

Lemma 7.5 Let E be a subset of Γ_{+}^{n} given by finitely many strict ℚ-linear inequalities as in (93), but with u = 0 (that is, only strict inequalities and no equalities appear in the definition of E). Then E is closed in Sper A_{x_{1}...x_{n}}.

Proof: Let the notation be as in (93) and let ω_{j} = (ω_{1j}, …, ω_{nj}), j ∈ {1, …, q}. Let the notation x^{ω_{j}} stand for x_{1}^{ω_{1j}} … x_{n}^{ω_{nj}}. Then we can write E as

E = {δ ∈ Sper A_{x_{1}...x_{n}} | x_{i} >_{δ} 0, i ∈ {1, …, n}, ν_{δ}(x^{ω_{j}}) > 0}.

For δ ∈ Sper A such that x_{i} >_{δ} 0, i ∈ {1, …, n}, saying that ν_{δ}(x^{ω_{j}}) > 0 is equivalent to saying that x^{−ω_{j}} is bounded below by every element of A. This is also equivalent to saying that x^{−ω_{j}} is bounded below by every monomial of the form Nx_{i}^{b}, N ∈ R, b ∈ ℕ, i ∈ {1, …, n}. For N ∈ R, b ∈ ℕ, i ∈ {1, …, n} and j ∈ {1, …, q}, let

S_{N,b,i,j} = {δ ∈ Sper A_{x_{1}...x_{n}} | x_{i} >_{δ} 0, i ∈ {1, …, n}, x^{−ω_{j}} ≥ Nx_{i}^{b} } .

Clearly, S_{N,b,i,j} is relatively closed in Sper A_{x_{1}...x_{n}}. We have

S_{E} = ⋂_{N,b,i,j} S_{N,b,i,j},  (96)

hence S_{E} is also closed. □

The point of the next two lemmas is to reduce Theorem 7.1 to the case when u = 0, that is, to the case when no ℚ-linear equalities appear in the definition of E, only strict inequalities.

Pick a subset J ⊂ {1, …, n}. We want to study blowings up of Sper A along the ideal (x_{J}). There are #J possible choices of coordinate charts, one for each element j ∈ J. For each j ∈ J, let π_{0j} : A → A_{j} denote the affine monomial blowing up, defined by J and j. Let E_{0j} ⊂ E be defined by

E_{0j} = {a ∈ E | π_{0j} is a blowing up with respect to a}

and let E′_{0j} denote the transform of E_{0j} under π_{0j}. By definitions, the set E′_{0j} is defined in E by imposing the additional inequalities a_{j} ≤ a_{q}, q ∈ J. As a subset of Γ_{+}^{n}, the set E′_{0j} is defined by the linear equalities and strict inequalities, which are the transforms in A_{j} of the equalities and inequalities (93) (the transforms of the non-strict inequalities a_{q} ≥ a_{j}, q ∈ J, which define E_{0j} inside E, have the form a′_{q} ≥ 0; these inequalities hold automatically for (a′_{1}, …, a′_{n}) ∈ Γ_{+}^{n} and need not be taken into account).

Lemma 7.6 Take two indices j, j̃ ∈ J. Assume that both E_{0j} and E_{0j̃} are not empty, in other words, that there exist a, ã ∈ E such that

a_{j} = min_{q∈J} a_{q} and  (97)
ãj̃ = min_{q∈J} ã_{q}.  (98)

Then there exists a chain of indices {j_{1}, …, j_{s}} ⊂ {1, …, n} such that j = j_{1}, j̃ = j_{s} and E_{0j_{q}} ∩ E_{0j_{q+1}} ≠ ∅ for all q ∈ {1, …, s − 1}.

Proof: Let Γ_{∘} denote the subset of Γ_{+} consisting of all the n-tuples of real numbers of the form (d, \underbrace{0, …, 0}_{n−1 zeroes}), d ∈ ℝ. Let Γ_{∘}^{n} denote the subset of Γ_{+}^{n} consisting of all the elements of the form

(c_{1}, …, c_{n}), c_{j} ∈ Γ_{∘}. Let E^{∘} = E ∩ Γ_{∘}^{n}, E_{0j}^{∘} = E_{0j} ∩ Γ_{∘}^{n}, etc.

Given an n-tuple ϵ = (ϵ_{1}, …, ϵ_{n}) of real numbers and an element d = (d_{1}, …, d_{n}) of Γ_{+} (where d_{q} ∈ ℝ), let d(ϵ) denote the element of Γ_{∘} defined by (d · ϵ, \underbrace{0, …, 0}_{n−1 zeroes}), where

d · ϵ = ∑_{q=1}^{n} d_{j}ϵ_{j}. If c = (c_{1}, …, c_{n}) is an element of Γ_{+}^{n}, with c_{j} ∈ Γ_{+}, we will write c(ϵ)

for (c_{1}(ϵ), c_{2}(ϵ), …, c_{n}(ϵ)) ∈ Γ_{∘}^{n}.

Take two indices l, l̃ ∈ J and elements c ∈ E_{0l}, c̃ ∈ E_{0l} ∩ E_{0l̃}. Since E_{0l} (resp. E_{0l} ∩ E_{0l̃}) is defined by linear equations and inequalities (not necessarily strict), we can choose real numbers

ϵ_{1} ≫ ϵ_{2} ≫ ⋯ ≫ ϵ_{n} > 0

such that c(ϵ) ∈ E_{0l} and c̃(ϵ) ∈ E_{0l} ∩ E_{0l̃}. Thus

E_{0l} ≠ ∅ ⟺ E_{0l}^{∘} ≠ ∅.  (99)

and

E_{0l} ∩ E_{0l̃} ≠ ∅ ⟺ E_{0l}^{∘} ∩ E_{0l̃}^{∘} ≠ ∅.  (100)

As a topological space, identify Γ_{∘}^{n} with ℝ^{n}, with the usual Euclidean topology. Since E^{∘} is defined in ℝ_{+}^{n} by linear equations and inequalities, it is connected in the Euclidean topology. Now, each E_{0l}^{∘} is relatively closed in E^{∘}. Hence the conclusion of the Lemma holds with E_{q} replaced by E_{q}^{∘}. In view of (99)–(100), this completes the proof of the Lemma. □

Next, we reduce the Theorem to the case when u = 0, that is, when E is defined by strict inequalities (and no equalities). Without loss of generality, we may assume that the equalities

∑_{i=1}^{n} θ_{il}a_{i} = 0, l ∈ {1, …, u}  (101)

appearing in the definition of E are linearly independent.

Lemma 7.7 If Theorem 7.1 is true for u = 0 (that is, in the case when there are no equalities in the definition of E) then it is true in general.

Proof: Assume Theorem 7.1 for u = 0. Apply Corollary 6.1 to the set S_{E}. For each i ∈ {1, …, s}, let

E_{i} = {a ∈ E | π_{i} is a sequence of blowings up with respect to a}  (102)

(cf. Remark 7.2). By Corollary 6.1 (2), E = ⋃_{i=1}^{s} E_{i}. Let E′_{i} denote the transform of E_{i}

under π_{i} (by definition of E_{i}, E′_{i} is well defined). By construction, E′_{i} is contained in the set {a_{r+1} = ⋯ = a_{n} = 0}, and is defined in Γ_{+}^{n} ∩ {a_{r+1} = ⋯ = a_{n} = 0} by finitely many strict

ℚ-linear inequalities — the transforms of the inequalities ∑_{q=1}^{n} ω_{qj}a_{q} > 0 under π_{i}. Let ι : Γ^{r} → Γ^{n}

denote the natural inclusion of the set {a_{r+1} = ⋯ = a_{n} = 0} in Γ^{n}. Let E_{i}^{†} = ι^{−1}(E′_{i}). Let p̃_{i} denote the natural projection p̃_{i} : Sper R[x_{i1}, …, x_{in}]_{x_{i1}...x_{in}} → Sper R[x_{i1}, …, x_{ir}]_{x_{i1}...x_{ir}}. The set S_{E′_{i}} can be written as

S_{E′_{i}} = {δ ∈ p̃_{i}^{−1} (E_{i}^{†}) | x_{ij} >_{δ} 0, ν_{δ}(x_{ij}) = 0 for r < j ≤ n} → Sper R[x_{i1}, …, x_{ir}]_{x_{i1}...x_{ir}}.

Now, S_{E_{i}^{†}} is connected by the u = 0 case of Theorem 7.1. Then S_{E′_{i}} is connected by Lemma 7.4. Hence S_{E_{i}} is connected by Corollary 5.1.

Let N_{i} denote the number of affine monomial blowings up composing π_{i} and let

N = max_{1≤i≤s}{N_{i}}.

To complete the proof of Lemma 7.7, we will now show that S_{E} is connected by induction on N. Let J ⊂ {1, …, n} be such that the first blowing up in each of the sequences π_{i} is a blowing up along J (such a J exists according to Remark 6.5).

The set E′_{0j} is defined in Γ_{+}^{n} by the linear equalities and strict inequalities, which are the transforms in A_{j} of the equalities and inequalities (93) (the transforms of the non-strict inequalities a_{q} ≥ a_{j}, q ∈ J, which define E_{0j} inside E, have the form a′_{q} ≥ 0; these inequalities hold automatically in Γ_{+}^{n} and need not be taken into account). Hence, by the induction assumption, S_{E′_{0j}} ≅ S_{E_{0j}} is connected.

Now, Lemma 7.6 shows that S_{E} = ⋃_{j∈J} S_{E_{0j}} is connected, as desired. This completes the

proof of Lemma 7.7. □

To complete the proof of Theorem 7.1, it remains to prove Proposition 7.1 Assume that u = 0 in (93), that is, E is defined by finitely many strict ℚlinear inequalities. Then S_{E} is connected.

First proof of Proposition 7.1: Write S_{E} = ⋂_{N,b,i,j} S_{N,b,i,j}, as in (96). Let

S_{N,b} = ⋂_{j=1}^{q} {δ ∈ ⋂_{i=1}^{n} S_{N,b,i,j} | x^{−ω_{j}} ≥ 1} .

For each N, b the set S_{N,b} is defined by finitely many monomial inequalities and the sign conditions x_{t} > 0, 1 ≤ t ≤ n. Thus S_{N,b} is connected by Lemma 4.1. If N ≤ N_{1} and b ≤ b_{1} then S_{N,b} ⊂ S_{N_{1},b_{1}}. Then the collection {S_{N,b} | N ∈ R, N > 0, b ∈ ℕ} is a filter of connected, closed subsets of Sper A_{x_{1}...x_{n}}. Then S_{E} is connected by Lemma 7.1. This completes the first proof of Proposition 7.1 and hence also of Theorem 7.1. □

Before giving the second proof of Theorem 7.1, we first prove two connectedness results.

Proposition 7.2 (fiber connectedness) Fix an element a = (a_{1}, …, a_{n}) ∈ Γ_{+}^{n}. The set S_{a} is connected in Sper(A).

Proof of Proposition 7.2: Let r = rat.rk a. Take a subset J ⊂ {1, …, n} and let

x_{J} = {x_{q} | q ∈ J}.

Let j ∈ J be such that

a_{j} = min{a_{q}}_{q∈J}  (103)

and consider the affine monomial blowing up

π : A → A′ = R[x′],

determined by J and j:

x′_{q} = x_{q}, a′_{q} = a_{q} if q∈/J or q = j

x′_{q} = \frac{x_{q}}{x_{j}}, a′_{q} = a_{q} − a_{j} if q ∈ J \ \{j\}.

Remark 7.2 By (103), π is an affine monomial blowing up with respect to δ for all δ ∈ S_{a}. In other words, the property of being an affine monomial blowing up with respect to δ depends only on a, not on the particular choice of δ ∈ S_{a}. Here and below, we will sometimes say that π is an affine monomial blowing up with respect to a.

Let (G′, a′_{1}, …, a′_{n}) be the ordered group generated by the a′_{i} (we have G′ = G) and let S_{a′} be the corresponding subset of Sper(A′). Then

S_{a} ≅ S_{a′}  (104)

by Proposition 5.1.

We now iterate the above procedure. By Corollaries 6.1 and 6.2 (where we take S = S_{a}), after a succession of such transformations we may assume (after passing to the new coordinates) that a_{1}, …, a_{r} are ℚ-linearly independent and

a_{r+1} = ⋯ = a_{n} = 0.  (105)

Again, note that by Remark 7.2, (104) and induction on the number of blowings up required, there is a single sequence of blowings up satisfying the conclusions of Corollary 6.1 which is a sequence of affine monomial blowings up simultaneously for all δ ∈ S_{a}. As a_{1}, …, a_{r} are ℚ-linearly independent, there exists a unique point

α = (t^{a_{1}}, …, t^{a_{r}}) ∈ Sper R[x_{1}, …, x_{r}]_{x_{1}...x_{r}}

such that

x_{i} >_{α} 0  (106)

and ν_{α}(x_{i}) = a_{i} for i ∈ {1, …, r}: because of the linear independence of the a_{i}, the support of α is the zero ideal of R[x_{1}, …, x_{r}] and the ordering ≤_{α} of R(x_{1}, …, x_{r}) is completely described by the inequalities (106) and

cx^{γ} < x^{ϵ} for all positive c ∈ R ⟺ ∑ γ_{j}a_{j} > ∑ ϵ_{j}a_{j}.  (107)

The fact that a_{1}, …, a_{r} are ℚ-linearly independent implies that (107) imposes a total ordering on the set of monomials and hence on R(x_{1}, …, x_{r}). The point α is closed in the relative topology of Sper R[x_{1}, …, x_{r}]_{x_{1}...x_{r}} because α has no non-trivial specializations. Now, let p̃ denote the natural projection p̃: Sper A_{x_{1},...,x_{r}} → Sper R[x_{1}, …, x_{r}]_{x_{1}...x_{r}}. Then

S_{a} = {δ ∈ p̃^{−1}({α}) | x_{q} > 0, ν_{δ}(x_{q}) = 0 for r < q ≤ n} ,

hence S_{a} is connected by Lemma 7.4. This completes the proof of Lemma 7.2. □

Corollary 7.1 Let S^{†} be a connected component of S_{E} and a an element of Γ_{+}^{n}. Then either S_{a} ⊂ S^{†} or S_{a} ∩ S^{†} = ∅. There exists a subset E^{†} ⊂ E such that S^{†} = S_{E^{†}}.

Take c = (c_{1}, …, c_{n}), d = (d_{1}, …, d_{n}) ∈ E such that c_{j} = d_{j} for j ∈ {1, …, n − 1}. We define the segment [c, d] ⊂ E as

[c, d] = {( c_{1} ; ⋮ ; c_{n−1} ; e_{n} ) | c_{n} ≤ e_{n} ≤ d_{n}}.  (108)

Let

S_{[c,d]} = {α ∈ S_{(c_{1},...,c_{n−1},e_{n})} | (c_{1}, …, c_{n−1}, e_{n}) ∈ [c, d]} .  (109)

For an element c = (c_{1}, …, c_{n}) ∈ Γ^{n}, we will write c_{j} = (c_{j1}, …, c_{jn}) with c_{ji} ∈ ℝ. In this way, we may think of c as an (n × n) matrix whose j-th row is c_{j}.

Proposition 7.3 Take c, d ∈ E such that :

(1) c_{j} = d_{j} for j ∈ {1, …, n − 1}

(2) c_{n1} < d_{n1}

(3) c_{11}, …, c_{n1}, d_{n1} are ℚ-linearly independent. Then S_{[c,d]} is connected.

Proof of Proposition 7.3: We start with two Lemmas.

Consider an element b ∈ Γ_{+}^{n} such that b_{1}, …, b_{n} are ℚ-linearly independent. Then S_{b} consists of a single point δ(b).

Lemma 7.8 Let U be a basic open set for the spectral topology of Sper A, containing δ(b). There exists a subset V ⊂ Γ_{+}^{n}, defined by strict ℚ-linear inequalities, such that b ∈ V and for any a ∈ V, we have S_{a} ⊂ U.

Proof: Let U = {α | f_{1}(α) > 0, …, f_{s}(α) > 0}. Write

f_{j} = ∑_{γ∈ℕ^{n}} c_{jγ}x^{γ}  (110)

Let M(j) = c_{j,γ(j)}x^{γ(j)} be the monomial of f_{j} of smallest valuation (which exists because b_{1}, …, b_{n} are ℚ-linearly independent).

For any j, 1 ≤ j ≤ s,

ν_{δ(b)} (x^{γ(j)}) < ν_{δ(b)} (x^{γ})  (111)

for all γ such that c_{jγ} ≠ 0. Writing γ = (γ_{1}, …, γ_{n}), γ(j) = (γ_{1}(j), …, γ_{n}(j)), (111) is equivalent

to saying that ∑_{q=1}^{n}(γ_{q} − γ_{q}(j))b_{q} > 0. Let V be the subset of Γ_{+}^{n}, defined by the inequalities

∑_{q=1}^{n}(γ_{q} − γ_{q}(j))a_{q} > 0 for all γ with c_{jγ} ≠ 0. Take an element a ∈ V and a point δ ∈ S_{a}. By

construction, each f_{j} has the same dominant monomial at δ as at δ(b). Then S_{a} ⊂ U, as desired. □

Let s, r ∈ {1, …, n − 1}, s ≤ r. Consider an element b ∈ Γ_{+}^{n} such that

b_{r+1} = ⋯ = b_{n} = 0

and b_{1}, …, b_{r} are ℚ-linearly independent. For 1 ≤ j ≤ r, write b_{j} = (b_{j1}, …, b_{jn}) with b_{jq} ∈ ℝ. Consider the (r × n)-matrix (b_{jq})_{1≤q≤n}^{1≤j≤r}. Assume that the rows of the (r × s)-matrix (b_{jq})_{1≤q≤s}^{1≤j≤r} are ℚ-linearly independent. Define b^{•} = (b_{1}^{•}, …, b_{n}^{•}) by b_{i}^{•} = b_{i}, i ≠ r + 1 and

b_{r+1}^{•} = (\underbrace{0, …, 0}_{s}, 1, 0, …, 0).

Lemma 7.9 We have \overline{S_{b}} ∩ S_{b^{•}} ≠ ∅.

Proof: Pick a point in S_{b^{•}}, for example,

α^{•} = (t^{b_{1}}, …, t^{b_{r}}, t^{b_{r+1}^{•}}, 1, …, 1)

(here we are representing α^{•} by a parametrized semi-curvette in R^{n} - see §3). Let

U = {α | f_{1}(α) > 0, …, f_{q}(α) > 0}

be a basic open set such that U ∋ α^{•}. It remains to prove that U ∩ S_{b} ≠ ∅. Let \overline{x} = (x_{1}, …, x_{r}) and \overline{\overline{x}} = (x_{r+2}, …, x_{n}). Write each f_{j} as a polynomial in the variables \overline{x} and x_{r+1}, whose coefficients are polynomials in \overline{\overline{x}}:

f_{j} = ∑_{γ=(\overline{γ},γ_{r+1})} c_{jγ}(\overline{\overline{x}})\overline{x}\overline{γ}x_{r+1}^{γ_{r+1}}  (112)

(here c_{jγ}(\overline{\overline{x}}) is the coefficient in f_{j} of the monomial \overline{x}\overline{γ}x_{r+1}^{γ_{r+1}}). Let M(j) be the term of smallest valuation among those appearing on the right hand side of (112); such a term is unique because b_{1}^{•}, …, b_{r+1}^{•} are ℚ-linearly independent. Let γ_{r+1}(j) (resp. \overline{γ}(j)) denote the exponent of x_{r+1} (resp. of \overline{x}) in M(j) and let c_{j,γ(j)} ∈ R[\overline{\overline{x}}] denote the coefficient of \overline{x}^{\overline{γ}(j)}x_{r+1}^{γ_{r+1}(j)} in (112), so that M(j) = c_{j,γ(j)}\overline{x}^{\overline{γ}(j)}x_{r+1}^{γ_{r+1}(j)}.

To say that M(j) is the monomial of f_{j} of smallest valuation means that for each j, 1 ≤ j ≤ q,

ν (\overline{x}^{\overline{γ}(j)}) ≤ ν (\overline{x}\overline{γ})  (113)

for all γ = (\overline{γ}, γ_{r+1}) such that c_{jγ} ≠ 0 and if equality holds in (113) then

γ_{r+1}(j) < γ_{r+1}.  (114)

Take ϵ ∈ R sufficiently small so that for all j ∈ {1, …, q} we have

|c_{j,γ(j)}(1, …, 1)ϵ^{γ_{r+1}(j)}| > ∑ ; \begin{matrix}γ = (\overline{γ}, γ_{r+1}) \\ γ_{r+1} > γ_{r+1}(j) \end{matrix} |c_{j,(\overline{γ}(j),γ_{r+1})}(1, …, 1)ϵ^{γ_{r+1}} | .

Now let α = (t^{b_{1}}, …, t^{b_{r}}, ϵ, 1, …, 1). We have α ∈ U ∩ S_{b}, as desired. □

We are now in the position to finish the proof of Proposition 7.3. Suppose S_{[c,d]} were not connected. By Corollary 7.1 we can write

[c, d] = B_{1} ∐ B_{2}  (115)

and

S_{[c,d]} = X_{1} ∐ X_{2},  (116)

such that X_{1} = S_{B_{1}} and X_{2} = S_{B_{2}} are open and closed in the relative topology of S_{[c,d]}. In view of assumption (1) of Proposition 7.3, (115) induces a decomposition of the segment [c_{n}, d_{n}] = B_{1n} ∐ B_{2n}.

Consider the segment [c_{n1}, d_{n1}] of the real line.

Lemma 7.10 Every point e_{n1} ∈ [c_{n1}, d_{n1}] can be covered by a set V (1), open in the topology induced on [c_{n1}, d_{n1}] by the Euclidean topology of ℝ, having the following property. Let V be the subset of Γ_{+}^{n} defined by

V = {e ∈ [c, d] | e_{n1} ∈ V (1)}.  (117)

Then all of S_{e}, e ∈ V are contained in the same set X_{s}, s = 1, 2.

Lemma 7.10 implies Proposition 7.3. Indeed, assume Lemma 7.10 and consider the decomposition (116). By Lemma 7.10, (116) induces a decomposition [c_{n1}, d_{n1}] = W_{1} ∐ W_{2} into two disjoint open sets in the Euclidean topology of ℝ, which gives the desired contradiction since [c_{n1}, d_{n1}] is connected.

It remains to prove Lemma 7.10. Proof of Lemma 7.10:

\underline{Case 1} : e_{n1}∈/ ∑_{j=1}^{n−1} ℚc_{j1}. Let e_{n} = (e_{n1}, 0, …, 0) and e = (c_{1}, …, c_{n−1}, e_{n}). Without loss of

generality, we may assume that S_{e} ⊂ X_{1} (the set S_{e} consists of a single point). Take a basic open set U of Sper A, containing S_{e} and disjoint from X_{2}. The existence of V (1) with the desired properties follows immediately from Lemma 7.8.

\underline{Case 2} : e_{n1} ∈ ∑_{j=1}^{n−1} ℚc_{j1}.

Let (e_{n2}, …, e_{nn}) be the unique (n − 1)-tuple of real numbers such that letting

e_{n} = (e_{n1}, e_{n2}, …, e_{nn}),

we have e_{n} ∈ ∑_{j=1}^{n−1} ℚc_{j}. Let π : A → A′ be a sequence of affine monomial blowings up with respect

to e, constructed in Proposition 6.1 and Corollary 6.1, such that, denoting by e′ the transform of e by π, the real numbers e′_{11}, …, e′_{n−1,1} are ℚ-linearly independent and

e′_{n} = 0.

Claim. Assuming e_{n} ≠ d_{n}, we can choose π to be a sequence of blowings up with respect to a small half-open interval of the form I = [e, h) ⊂ [c, d] with h_{n1} > e_{n1}; in other words, for any b ∈ [e, h), and any α ∈ S_{b}, we have that ν_{α} is non-negative on A′. Proof of Claim: We will go through the procedure of Proposition 6.1 step by step and show by induction on the number of blowings up composing the sequence π that there exists a choice of π as required in the Claim. Recall (Remark 6.4) that the algorithm of Proposition 6.1 consists of choosing the center of blowing up at each step. The algorithm works for every choice of coordinate chart at each step, but the choice of the coordinate chart determines whether the given blowing up is a blowing up with respect to a certain valuation ν.

Let J be the subset of {1, …, n} prescribed by the algorithm of Proposition 6.1 so that the first blowing up π_{1} appearing in π is a blowing up along (x_{J}). In the discussion that follows,

recall that c_{i} = e_{i} for i ∈ {1, …, n − 1}. Since dim_{ℚ} ∑_{i=1}^{n} ℚe_{i1} = dim_{ℚ} ∑_{i=1}^{n} ℚe_{i} = n − 1, either

there exists a unique j ∈ J such that e_{j1} < e_{q1}, q ∈ J \ {j}, or there exist two indices j, l ∈ J such that e_{l} = e_{j} < e_{q}, q ∈ J \ {l, j}. In the first case, we have

b_{j} < b_{q}, q ∈ J \ {j} for every b = {b_{1}, …, b_{n}} ∈ [e, h)  (118)

assuming that h_{n1} − e_{n1} is a sufficiently small positive real number. Use j to determine the coordinate chart for π_{1}; then by (118) π_{1} is also a blowing up with respect to every b ∈ [e, h), as desired.

Next, assume that e_{l} = e_{j} are the minimal elements of e_{J}. Take h ∈ [e, d] with h_{n1} − e_{n1} > 0 sufficiently small so that

b_{l}, b_{j} < b_{q}, q ∈ J \ {l, j} for every b = {b_{1}, …, b_{n}} ∈ [e, h).  (119)

We have either b_{j} ≤ b_{l} for all b ∈ [e, h) or b_{j} ≥ b_{l} for all b ∈ [e, h) (or both). If b_{j} ≤ b_{l}, let j determine the choice of coordinate chart for π_{1}; if b_{j} ≥ b_{l} for b ∈ [e, h), let the coordinate chart for π_{1} be determined by l. In either case, π_{1} is a blowing up with respect to b for all b ∈ [e, h). Repeating the above procedure at every step of the algorithm of Proposition 6.1, we can choose π as required in the Claim. The Claim is proved.

For any b ∈ [e, h), we have S_{b′} ≅ S_{b} by Proposition 5.1. Let I′ denote the transform of I under π, defined in the obvious way.

Since the natural map π^{∗} : Sper A′ → Sper A is continuous and preserves rational rank, we will work with A′, c′, d′, e′, I′ instead of A, c, d, e, I.

The decomposition (115) induces a decomposition of I and hence also a decomposition

I′ = B′_{1} ∐ B′_{2},

such that S_{B′_{1}} and S_{B′_{2}} are open and closed in the induced topology of S_{I′}.

Let e_{n}^{•} = (0, 1, 0, …, 0), e^{•} = (e′_{1}, …, e′_{n−1}, e_{n}^{•}). Without loss of generality, assume that e^{•} ∈ B′_{1}. Let U be a basic open set of Sper A having non-empty intersection with S_{e^{•}} and disjoint from S_{B′_{2}}. Now Lemma 7.9 (with r = n − 1 and s = 1) shows that S_{e^{•}} ∩ \overline{S_{e′}} ≠ ∅ and Lemma 7.8 shows that for a sufficiently small interval V^{†}(1) = (0, ϵ) of the real line, ϵ > 0, if b′_{n1} ∈ V^{†}(1) then S_{b′} ⊂ U. Let V_{+}(1) = [0, ϵ) and let V_{+} = {e′ ∈ [c′, d′] | e′_{n1} ∈ V_{+}(1)}. This

proves that ⋃_{b′∈V_{+}} S_{b′} ⊂ S_{B′_{1}}.

Coming back to the original interval [c_{n1}, d_{n1}] (that is, before performing the blowing up sequence π), we have shown that all the S_{b} for b ∈ [e, h) lie in the same set X_{s}, s ∈ {1, 2}, provided we take h_{n1} sufficiently close to e_{n1} and to the right of e_{n1}. Now, assuming e_{n} ≠ c_{n}, let J be a small half-open interval of the form (v, e] with v < e. Repeating the above reasoning with I replaced by J, we obtain that for V = (v, h), ⋃_{b′∈V} S_{b′} ⊂ S_{B′_{1}}, so the open interval

V (1) = (v_{n1}, h_{n1}) satisfies the conclusion of the Lemma. This completes the proof of Lemma 7.10 and Proposition 7.3. □

Second proof of Proposition 7.1: The main idea of this proof is the following. Take two points α and β in S_{E}. Let

a = (a_{1}, …, a_{n}) : = (ν_{α}(x_{1}), …, ν_{α}(x_{n})) and  (120)
b = (b_{1}, …, b_{n}) : = (ν_{β}(x_{1}), …, ν_{β}(x_{n})).  (121)

We join α and β by a “staircase” in Γ^{n} where each stair lies entirely in a connected component of S_{E}. Two examples at the end of the paper show that we have to do this with some care.

Let r ∈ {1, …, n − 1}. Consider an element b ∈ Γ_{+}^{n} such that b_{r+1} = ⋯ = b_{n} = 0 and b_{1}, …, b_{r} are ℚ-linearly independent. For 1 ≤ j ≤ r, write b_{j} = (b_{j1}, …, b_{jn}) with b_{jq} ∈ ℝ. Consider the (r × n)-matrix (b_{jq})_{1≤q≤n}^{1≤j≤r}.

Lemma 7.11 There exists b′ = (b′_{1}, …, b′_{n}) ∈ Γ_{+}^{n} having the following properties: (1) b ∼_{∘} b′ (in particular, b′_{r+1} = ⋯ = b′_{n} = 0)

(2) writing b′_{j} = (b′_{j1}, …, b′_{jn}), we have that the rows of the (r × r)-matrix (b′_{jq})_{1≤q≤r}^{1≤j≤r} are ℚ-linearly independent.

Proof: We will gradually replace b by elements of Γ_{+}^{n}, isomorphic to b in the sense of OGM(n) until we reach an element of Γ_{+}^{n} satisfying the conclusion of the Lemma.

Let z be the number of non-zero columns in the matrix (b_{jq})_{1≤q≤n}^{1≤j≤r}:

z = #{q ∈ {1, …, n} | ∃j, 1 ≤ j ≤ r, such that b_{jq} ≠ 0}.

Let ℓ be the greatest index in {1, …, r} such that the column vectors

( b_{11} ; b_{21} ; ⋮ ; b_{r1} ) , …, ( b_{1ℓ} ; b_{2ℓ} ; ⋮ ; b_{rℓ} )

are ℝ-linearly independent. We proceed by induction on z − ℓ. First, suppose z − ℓ = 0. Since ℓ = z ≤ r, the n-rows b_{1}, …, b_{n} are obtained from the rows of the matrix (b_{jq})_{1≤q≤r}^{1≤j≤r} by adding n − r zeroes at the end. Since b_{1}, …, b_{r} are ℚ-linearly independent by assumption, so are the rows of (b_{jq})_{1≤q≤r}^{1≤j≤r}. Thus in the case z − ℓ = 0 we may take b′ = b and there is nothing more to prove.

Assume ℓ < z. Permute the columns of (b_{jq})_{1≤q≤n}^{1≤j≤r} by moving all the zero columns to the right of the matrix; this does not change the equivalence class of b with respect to the relation ∼_{∘}. If this permutation increases the number ℓ, the proof is finished by induction on z − ℓ. Suppose the number ℓ stays constant after the above permutation of columns. This means that, after the above permutation of columnes,

( b_{1,ℓ+1} ; b_{2,ℓ+1} ; ⋮ ; b_{r,ℓ+1} ) is an ℝ-linear combination of ( b_{1j} ; b_{2j} ; ⋮ ; b_{rj} ) , 1 ≤ j ≤ ℓ.

Replace b_{j,ℓ+1} by 0 for 1 ≤ j ≤ r. This operation does not change the equivalence class of b, but decreases the number z of non-zero columns by 1. The result follows by induction on z − ℓ. □

Lemma 7.12 For any connected component S_{E^{†}} of S_{E}, there exists b ∈ E^{†} such that

rat.rk(b) = n.

Proof: Take b ∈ E^{†} such that rat.rk(b) = max {rat.rk(b^{†}) | b^{†} ∈ E^{†} }. Suppose

rat.rk(b) = r < n.

We will now construct b^{•} ∈ Γ_{+}^{n} such that

S_{b^{•}} ∩ \overline{S_{b}} ≠ ∅  (122)

and

rat.rk(b^{•}) = r + 1.  (123)

This will contradict the maximality of r.

Let A → A′ be a sequence of affine monomial blowings up with respect to b, constructed in §6, such that, denoting by b′ the transform of b by π, b′_{1}, …, b′_{r} are ℚ-linearly independent and

b′_{r+1} = ⋯ = b′_{n} = 0.

Since the natural map π^{∗} : Sper A → Sper A′ is continuous and preserves rational rank, we may replace A by A′ and b by b′: if the conditions (122) and (123) are satisfied in Sper A′, they will still be satisfied after applying π^{∗} to everything in sight. From now on, we drop the primes and assume that b_{1}, …, b_{r} are ℚ-linearly independent and b_{r+1} = ⋯ = b_{n} = 0.

Define b^{•} = (b_{1}^{•}, …, b_{n}^{•}) by b_{i}^{•} = b_{i}, i ≠ r + 1 and b_{r+1}^{•} = (\underbrace{0, …, 0}_{r}, 1, 0, …, 0). By Lemma

7.9, (122) holds. Since b ∈ E, we have

b^{•} ∈ E  (124)

because, by the ℚ-linear independence of the rows of the matrix (b_{jq})_{1≤q≤r}^{1≤j≤r}, the presence of b_{r+1}^{•} does not affect the strict inequalities defining E: any integer multiple of b_{r+1}^{•} is infinitesimal in absolute value compared to any non trivial ℤ-linear combination of b_{1}, …, b_{r}. Since S_{b} ⊂ E^{†},

(122) implies that S_{b^{•}} ⊂ E^{†}. Since rat.rk(b^{•}) = r + 1, this contradicts the maximality of r. This completes the proof of the Lemma. □

For a = (a_{1}, …, a_{n}), write a_{j} = (a_{j1}, …, a_{jn}). Take a ∈ E such that a_{11}, …, a_{n1} are ℚ-linearly independent, in particular, rat.rk a = n (such an a exists because the n-tuples (a_{11}, …, a_{n1}) such that a_{11}, …, a_{n1} are ℚ-linearly independent are dense in ℝ^{n}).

Now suppose that S_{E} is not connected and let S^{(1)}, S^{(2)} be two open and closed sets such that S_{E} = S^{(1)} ∐ S^{(2)} with S^{(1)} containing S_{a}. Let E^{(1)}, E^{(2)} be the subsets of E such that S^{(1)} = S_{E^{(1)}} and S^{(2)} = S_{E^{(2)}}; E^{(1)} and E^{(2)} exist by Corollary 7.1.

Lemma 7.13 There exists b ∈ E^{(2)} such that a_{11}, …, a_{n1}, b_{11}, …, b_{n1} are ℚ-linearly independent.

Proof: According to the preceding lemma, we can find c ∈ E^{(2)} such that rat.rk(c) = n. Then S_{c} consists of a single point. Let U be a basic open set containing S_{c} and such that

U ∩ S^{(1)} = ∅.  (125)

Let V ⊂ Γ_{+}^{n} be a set satisfying the conclusion of Lemma 7.8. Since the n-tuples (b_{11}, …, b_{n1}) such that (a_{11}, …, a_{n1}, b_{11}, …, b_{n1}) are ℚ-linearly independent are dense in R^{n}, there exists b ∈ E ∩ V such that (a_{11}, …, a_{n1}, b_{11}, …, b_{n1}) are ℚ-linearly independent. By Lemma 7.8, S_{b} ⊂ U. By (125), b ∈ E^{(2)}, as desired. □

Let a ∈ E^{(1)}, b ∈ E^{(2)} be such that a_{11}, …, a_{n1}, b_{11}, …, b_{n1} are ℚ-linearly independent.

Remark 7.3 If λ, μ ∈ ℚ, λ ≠ μ, then λa_{11} +(1−λ)b_{11}, …, λa_{n1} +(1−λ)b_{n1}, μa_{11} +(1−μ)b_{11}, …, μa_{n1} + (1 − μ)b_{n1} are ℚ-linearly independent.

In the sequel, let N be a large natural number and λ = \frac{i}{N} , μ = \frac{i + 1}{N}.

Lemma 7.14 For N ∈ ℕ sufficiently large we have

( (\frac{i}{N} a_{11} + (1 − \frac{i}{N} ) b_{11}, 0, …, 0) ; ⋮ ; (\frac{i}{N} a_{j−1,1} + (1 − \frac{i}{N} ) b_{j−1,1}, 0, …, 0) ; (\frac{i+1}{N}a_{j1} + (1 − \frac{i+1}{N}) b_{j1}, 0, …, 0) ; ⋮ ; (\frac{i+1}{N}a_{n1} + (1 − \frac{i+1}{N}) b_{n1}, 0, …, 0) ) ∈ E.

for all i ∈ {0, …, N − 1} and all j ∈ {1, …, n}.

Proof: Since a_{11}, …, a_{n1} are ℚ-linearly independent, saying that ∑_{i=1}^{n} ω_{il}a_{i} > 0, l ∈ {1, …, q} is

equivalent to saying that

∑_{i=1}^{n} ω_{il}a_{i1} > 0, l ∈ {1, …, q}.

Similarly, we have

∑_{i=1}^{n} ω_{il}b_{i1} > 0, l ∈ {1, …, q}.

The set of n-tuples (c_{11}, …, c_{n1}) ∈ ℝ^{n} such that ∑_{i=1}^{n} ω_{il}c_{i1} > 0, l ∈ {1, …, q}, is open and convex

and contains both (a_{11}, …, a_{n1}) and (b_{11}, …, b_{n1}). Then for N sufficiently large we have

∑_{i=1}^{j−1} ω_{il} (\frac{i}{N} a_{i1} + (1 − \frac{i}{N} ) b_{i1}) + ∑_{i=j}^{n} ω_{il} (\frac{i + 1}{N}a_{i1} + (1 − \frac{i + 1}{N}) b_{i1}) =

∑_{i=1}^{n} ω_{il} (\frac{i}{N} a_{i1} + (1 − \frac{i}{N} ) b_{i1}) + ∑_{i=j}^{n} ω_{il} (\frac{1}{N} a_{i1} − \frac{1}{N} b_{i1}) > 0,

l ∈ {1, …, q}. This completes the proof of the Lemma. □

For each i ∈ {0, …, N − 1} and j ∈ {1, …, n}, consider the pair

a(i, j − 1) := ( (\frac{i}{N} a_{11} + (1 − \frac{i}{N} ) b_{11}, 0, …, 0) ; ⋮ ; (\frac{i}{N} a_{j−1,1} + (1 − \frac{i}{N} ) b_{j−1,1}, 0, …, 0) ; (\frac{i+1}{N}a_{j1} + (1 − \frac{i+1}{N}) b_{j1}, 0, …, 0) ; ⋮ ; (\frac{i+1}{N}a_{n1} + (1 − \frac{i+1}{N}) b_{n1}, 0, …, 0) ) ,  (126)

a(i, j) := ( (\frac{i}{N} a_{11} + (1 − \frac{i}{N} ) b_{11}, 0, …, 0) ; ⋮ ; (\frac{i}{N} a_{j,1} + (1 − \frac{i}{N} ) b_{j,1}, 0, …, 0) ; (\frac{i+1}{N}a_{j+1,1} + (1 − \frac{i+1}{N}) b_{j+1,1}, 0, …, 0) ; ⋮ ; (\frac{i+1}{N}a_{n1} + (1 − \frac{i+1}{N}) b_{n1}, 0, …, 0) ) .  (127)

of elements of Γ_{+}^{n}.

We can now finish proving the connectedness of S_{E}. Take i ∈ {0, …, N − 1} and j ∈ {1, …, n}. Up to renumbering the components c_{1}, …, c_{n}, the pair of points c = a(i, j − 1) and d = a(i, j) satisfy the hypotheses of Proposition 7.3. Then Proposition 7.3 says that S_{a(i,j−1)} and S_{a(i,j)} are contained in the same connected component of S_{E}, so a(i, j −1) and a(i, j) belong to the same set E^{(s)}, s = 1, 2. Put

b_{trunc} := ( (b_{11}, 0, …, 0) ; ⋮ ; (b_{n1}, 0, …, 0) ) .

Since b ∈ E^{(2)} and S_{b} = S_{b_{trunc}} by linear independence of b_{11}, …, b_{n1}, we must have b_{trunc} ∈ E^{(2)}.

Since b_{trunc} = a(0, n) and a(i, j − 1) and a(i, j) belong to the same set E^{(s)} for all i, j, we have a(i, j) ∈ E^{(2)} for all i, j by induction on (i, n − j) in the lexicographical ordering (note that a(i, 0) = a(i + 1, n)). Then

a(N − 1, 0) = a_{trunc} := ( (a_{11}, 0, …, 0) ; ⋮ ; (a_{n1}, 0, …, 0) ) ∈ E^{(2)}

so, finally, a ∈ E^{(2)} which contradicts the fact that a was chosen to lie in E^{(1)}. This completes the second proof of Proposition 7.1 (the connectedness of S_{E}) and with it of Theorem 7.1.

Next, we explain how to use Theorem 7.1 to give another proof of Theorem 4.1. Let the notation be as in the proof of Theorem 4.1. Let C be the set of all δ ∈ Sper A such that the inequalities (34) are satisfied with α replaced by δ and x_{q}(δ) has the same sign as x_{q}(α) for all q such that x_{q}∈/ < α, β >. The set C contains both α and β and none of the polynomials g_{i} change sing on C. Thus to complete the proof of Theorem 4.1, it remains to prove

Corollary 7.2 (of Theorem 7.1). The set C is connected.

Proof: For an index q ≤ l, replacing x_{q} by −x_{q} does not change the problem. Thus we may assume that the sign conditions appearing in the definition of C are all of the form x_{q} > 0 for q ≤ l. We proceed by induction on n−l. Il l = n then the Corollary is a special case of Theorem 7.1. Assume that l < n and that the result is known for all the smaller values of n − l. Let C_{+} = {δ ∈ C | x_{n}(δ) > 0}, C_{−} = {δ ∈ C | x_{n}(δ) < 0} and C_{0} = {δ ∈ C | x_{n}(δ) = 0}. By the induction assumption, each of C_{+}, C_{−} and C_{0} is connected. Take any point δ ∈ C_{0} and a basic open set U of Sper A, containing δ. It is easy to show, by an argument similar to that of Lemma 7.9, that U ∩ C_{+} ≠ ∅ and U ∩ C_{−} ≠ ∅. Hence C_{0} ⊂ C̄_{+} and C_{0} ⊂ C̄_{−}. This proves that C = C_{+} ∪ C_{0} ∪ C_{−} is connected, as desired. □

The next two examples show that the intuition that “convex sets in Γ_{+}^{n} give rise to connected sets in Sper A” is not completely accurate.

Example 1: Let n = 2. Let c = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) , d = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ∈ Γ^{2}. Let S_{[c,d]} be the subset of

Sper A, defined in (108) and (109). Let S_{]c,d]} = S_{[c,d]} \ S_{c}. Then S_{[c,d]} is not connected: we have S_{[c,d]} = S_{c} ∐ S_{]c,d]} = S_{c} ∐ S_{d} and both S_{c} and S_{d} are open and closed. In particular, Proposition 7.3 is false without the assumption that c_{11}, …, c_{n1}, d_{n1} are linearly independent. Another way of interpreting this example is that intervals consisting only of rank 1 valuations do not give rise to connected sets in Sper A.

Example 2: In this example, higher rank valuations appear, but, again, c_{11}, …, c_{n1}, d_{n1} are ℚ-linearly dependent. Let n = 3 and take c_{1} = d_{1} = e_{1} = (1, 0, 0), c_{2} = d_{2} = e_{2} = (0, 1, 0), c_{3} = (0, 0, 0), d_{3} = (1, 0, 0), e_{3} = (e_{31}, e_{32}, e_{33}), c_{3} ≤ e_{3} ≤ d_{3}. Then c_{11}, c_{21}, c_{31}, d_{31} are ℚlinearly dependent and the segment [c_{31}, d_{31}] = [0, 1] is a disjoint union of V^{(1)} = {e_{31} = 0} and V^{(2)} = {0 < e_{31} ≤ 1}. The decomposition [c_{31}, d_{31}] = V^{(1)} ∐ V^{(2)} induces a decomposition S_{[c,d]} = S_{V^{(1)}} ∐ S_{V^{(2)}} into two disjoint sets. We claim that S_{V^{(1)}} and S_{V^{(2)}} are both open and closed in the induced topology of S_{[c,d]}. To see this, it is sufficient to cover S_{V^{(1)}} by basic open sets of Sper A, disjoint from S_{V^{(2)}}, and vice versa.

Let ℕ_{+} denote the set of strictly positive integers. For each q ∈ ℕ_{+}, let U_{q} denote the basic open set of Sper A, defined by the equation x_{3} − x_{2}^{q} > 0. For a point δ ∈ S_{V^{(2)}} we have qν_{δ}(x_{2}) = (0, q, 0) < (e_{31}, e_{32}, e_{33}) = ν_{δ}(x_{3}) (see the definition of the lexicographical ordering at the end of section §3), so x_{2}^{q} >_{δ} x_{3} and δ∈/U_{q}. On the other hand, if δ ∈ S_{V^{(1)}} then ν_{δ}(x_{3}) has the form (0, e_{32}, e_{33}). Taking q > e_{32}, we have qν_{δ}(x_{2}) = (0, q, 0) > (0, e_{32}, e_{33}) = ν_{δ}(x_{3}), so

x_{2}^{q} < x_{3} and δ ∈ U_{q}. Thus S_{V^{(1)}} ⊂ ⋃_{q∈ℕ_{+}} U_{q} and U_{q} ∩ S_{V^{(2)}} = ∅ for all q ∈ ℕ_{+}. This proves that

S_{V^{(1)}} is relatively open in the induced topology.

Next, let W_{q} denote the basic open set of Sper A, defined by the equation x_{1} − x_{3}^{q} > 0, q ∈ ℕ_{+}. For a point δ ∈ S_{V^{(1)}} we have qν_{δ}(x_{3}) = (0, qe_{32}, qe_{33}) < (1, 0, 0) = ν_{δ}(x_{1}), so x_{3}^{q} > x_{1} and δ∈/W_{q}. On the other hand, if δ ∈ S_{V^{(2)}} then ν_{δ}(x_{3}) has the form (e_{31}, e_{32}, e_{33}) with e_{31} > 0. Taking q > \frac{1}{e_{31}} , we have qν_{δ}(x_{3}) = (qe_{31}, qe_{32}, qe_{33}) > (1, 0, 0) = ν_{δ}(x_{1}), so x_{3}^{q} < x_{1} and δ ∈ W_{q}. Thus S_{V^{(2)}} ⊂ ⋃_{q∈ℕ_{+}} W_{q} and W_{q} ∩ S_{V^{(1)}} = ∅ for all q ∈ ℚ_{+}. This proves that S_{V^{(2)}} is relatively

open in the induced topology, as desired.

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