Decidability and Local-Global Principles,

by L. Darnière.

Contemporary Mathematics, 270 (2000) 145-167

Proceedings of the workshop: Hilbert's 10'th problem, Relations with Arithmetic and Algebraic Geometry.
University of Gent, November 2-5, 1999.

Short abstract

This survey paper attempts to give a comprehensive overview of the decidability results obtained in recent years for rings (and fields) based on the use of a geometric Local-Global Principle. Part I presents the deep theorems of algebraic geometry which provide many natural examples of rings satisfying a Local-Global Principle, in every characteristic. Part II is devoted to the numerous developments in model theory which may have been stimulated by the results of part I, and specially to decidability results for those rings.

Mathematics Subject Classification (2000)

Primary: 11U05, 11U09
Secondary: 11R04, 11G35

Electronic version of the paper

Version April 00 (23 pages): dvi, Postscript.