%0 Journal Article %T Approximate roots of a valuation and the Pierce-Birkhoff Conjecture %+ Laboratoire Angevin de Recherche en Mathématiques (LAREMA) %+ Department of Mathematics [Baton Rouge] (LSU Mathematics) %+ Institut de Mathématiques de Toulouse UMR5219 (IMT) %A Lucas, François %A Madden, James %A Schaub, Daniel %A Spivakovsky, Mark %< avec comité de lecture %@ 0240-2963 %J Annales de la Faculté des Sciences de Toulouse. Mathématiques. %I Université Paul Sabatier _ Cellule Mathdoc %V 21 %N 2 %P 259-342 %8 2012 %D 2012 %Z 1003.1188 %R 10.5802/afst.1336 %K Pierce-Birkhoff %K real spectrum %K valuation %K approximate root %K piecewise polynomial %K blowing up %K complete ideal %Z 14P10 %Z Mathematics [math]/Algebraic Geometry [math.AG]Journal articles %X 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. %G English %2 https://ujm.hal.science/ujm-00461549v3/document %2 https://ujm.hal.science/ujm-00461549v3/file/approximatelastversion.pdf %L ujm-00461549 %U https://ujm.hal.science/ujm-00461549 %~ UNIV-ST-ETIENNE %~ UNIV-TLSE2 %~ UNIV-TLSE3 %~ CNRS %~ UNIV-ANGERS %~ INSA-TOULOUSE %~ LAREMA %~ FMPL %~ INSMI %~ OPENAIRE %~ IMT %~ UT1-CAPITOLE %~ INSA-GROUPE %~ UDL %~ UNIV-UT3 %~ UT3-INP %~ UT3-TOULOUSEINP