%0 Journal Article %T On the Pierce-Birkhoff Conjecture %+ Laboratoire Angevin de Recherche en Mathématiques (LAREMA) %+ Institut de Mathématiques de Toulouse UMR5219 (IMT) %A Lucas, François %A Schaub, Daniel %A Spivakovsky, Mark %< avec comité de lecture %@ 0021-8693 %J Journal of Algebra %I Elsevier %V vol. 435 %P p. 124-158 %8 2015-08-01 %D 2015 %Z 1207.6463 %R 10.1016/j.jalgebra.2015.04.005 %K real spectrum %K approximate root %K valuation %K blowing up %K complete ideal %K piecewise polynomial %K Pierce-Birkhoff %Z 14P10 %Z Mathematics [math]/Algebraic Geometry [math.AG]Journal articles %X This paper represents a step in our program towards the proof of the Pierce--Birkhoff conjecture. In the nineteen eighties J. Madden proved that the Pierce-Birkhoff conjecture for a ring A$is equivalent to a statement about an arbitrary pair of points $\alpha,\beta\in\sper\ A$ and their separating ideal $<\alpha,\beta>$; we refer to this statement as the Local Pierce-Birkhoff conjecture at $\alpha,\beta$. In this paper, for each pair $(\alpha,\beta)$ with $ht(<\alpha,\beta>)=\dim A$, we define a natural number, called complexity of $(\alpha,\beta)$. Complexity 0 corresponds to the case when one of the points $\alpha,\beta$ is monomial; this case was already settled in all dimensions in a preceding paper. Here we introduce a new conjecture, called the Strong Connectedness conjecture, and prove that the strong connectedness conjecture in dimension n-1 implies the connectedness conjecture in dimension n in the case when $ht(<\alpha,\beta>)$ is less than n-1. We prove the Strong Connectedness conjecture in dimension 2, which gives the Connectedness and the Pierce--Birkhoff conjectures in any dimension in the case when $ht(<\alpha,\beta>)$ less than 2. Finally, we prove the Connectedness (and hence also the Pierce--Birkhoff) conjecture in the case when dimension of A is equal to $ht(<\alpha,\beta>)=3$, the pair $(\alpha,\beta)$ is of complexity 1 and $A$ is excellent with residue field the field of real numbers. %G English %2 https://ujm.hal.science/ujm-00721187/document %2 https://ujm.hal.science/ujm-00721187/file/pfPB3.pdf %L ujm-00721187 %U https://ujm.hal.science/ujm-00721187 %~ UNIV-ST-ETIENNE %~ UNIV-NANTES %~ UNIV-TLSE2 %~ UNIV-TLSE3 %~ CNRS %~ UNIV-ANGERS %~ INSA-TOULOUSE %~ LAREMA %~ FMPL %~ INSMI %~ IMT %~ UT1-CAPITOLE %~ CHL %~ INSA-GROUPE %~ UDL %~ UNIV-UT3 %~ UT3-INP %~ UT3-TOULOUSEINP