Approximate roots of a valuation and the Pierce-Birkhoff Conjecture

François Lucas 1 James Madden 2 Daniel Schaub 1 Mark Spivakovsky 3
1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
2 LSU Mathematics - Department of Mathematics [Baton Rouge]
3 IMT - Institut de Mathématiques de Toulouse UMR5219
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.
Keywords : Pierce-Birkhoff real spectrum valuation approximate root piecewise polynomial blowing up complete ideal
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Journal articles
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https://hal-ujm.archives-ouvertes.fr/ujm-00461549
Contributor : Daniel Schaub <>
Submitted on : Thursday, February 9, 2012 - 5:11:47 PM
Last modification on : Friday, January 10, 2020 - 9:08:59 PM
Long-term archiving on: Thursday, May 10, 2012 - 2:58:02 AM

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François Lucas, James Madden, Daniel Schaub, Mark Spivakovsky. Approximate roots of a valuation and the Pierce-Birkhoff Conjecture. Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc 2012, 21 (2), pp.259-342. ⟨10.5802/afst.1336⟩. ⟨ujm-00461549v3⟩

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