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

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

Domains: Algebraic Geometry [math.AG]

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., 2012, 21 (2), pp.259-342. ⟨10.5802/afst.1336⟩. ⟨ujm-00461549v3⟩