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.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas
Contributor : Daniel Schaub <>
Submitted on : Thursday, March 4, 2010 - 11:48:38 PM
Last modification on : Friday, January 10, 2020 - 9:08:06 PM
Long-term archiving on: Friday, June 18, 2010 - 10:19:05 PM


Files produced by the author(s)


  • HAL Id : ujm-00461549, version 1
  • ARXIV : 1003.1188


François Lucas, James Madden, Daniel Schaub, Mark Spivakovsky. Approximate roots of a valuation and the Pierce-Birkhoff Conjecture. 2010. ⟨ujm-00461549v1⟩



Record views


Files downloads