Approximate roots of a valuation and the Pierce-Birkhoff Conjecture

TL;DR

This paper advances the proof of the Pierce-Birkhoff Conjecture for regular rings by introducing approximate roots of valuations, proving the conjecture in dimension 2, and connecting it to the Connectedness conjectures.

Contribution

It introduces the notion of approximate roots of valuations and proves the Pierce-Birkhoff Conjecture for 2-dimensional regular rings.

Findings

Proves Pierce-Birkhoff conjecture for 2-dimensional regular rings.

Introduces explicit formulas for sets in the real spectrum using approximate roots.

Establishes connections between connectedness conjectures 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.