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.
Keywords
Cite
@article{arxiv.1003.1188,
title = {Approximate roots of a valuation and the Pierce-Birkhoff Conjecture},
author = {François Lucas and James Madden and Daniel Schaub and Mark Spivakovsky},
journal= {arXiv preprint arXiv:1003.1188},
year = {2012}
}