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Université Jean Monnet – Saint-Etienne |  |
Approximate roots of a valuation and the Pierce-Birkhoff Conjecture
Abstract : This paper is a step in our program for proving the Piece-Birkhoff Conjecture for regular rings of any dimension (this would contain, in particular, the classical Pierce-Birkhoff conjecture which deals with polynomial rings over a real closed field). We first recall the Connectedness and the Definable Connectedness conjectures, both of which imply the Pierce - Birkhoff conjecture. Then we introduce the notion of a system of approximate roots of a valuation v on a ring A (that is, a collection Q of elements of A such that every v-ideal is generated by products of elements of Q). We use approximate roots to give explicit formulae for sets in the real spectrum of A which we strongly believe to satisfy the conclusion of the Definable Connectedness conjecture. We prove this claim in the special case of dimension 2. This proves the Pierce-Birkhoff conjecture for arbitrary regular 2-dimensional rings.
https://hal-ujm.archives-ouvertes.fr/ujm-00461549
Contributor : Daniel Schaub <>
Submitted on : Thursday, February 9, 2012 - 5:11:47 PM
Last modification on : Monday, March 9, 2020 - 6:15:59 PM
Document(s) archivé(s) le : Thursday, May 10, 2012 - 2:58:02 AM
Contributor : Daniel Schaub <>
Submitted on : Thursday, February 9, 2012 - 5:11:47 PM
Last modification on : Monday, March 9, 2020 - 6:15:59 PM
Document(s) archivé(s) le : Thursday, May 10, 2012 - 2:58:02 AM
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