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arXiv · 1003.1188

Approximate roots of a valuation and the Pierce-Birkhoff Conjecture

François Lucas·James Madden·Daniel Schaub·Mark Spivakovsky
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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.

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François Lucas, James Madden, Daniel Schaub, Mark Spivakovsky. 2012-02-09. Approximate roots of a valuation and the Pierce-Birkhoff Conjecture. https://arxiv.org/abs/1003.1188

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