On the Pierce-Birkhoff Conjecture

Abstract : 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.
Type de document :
Pré-publication, Document de travail
2012
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https://hal-ujm.archives-ouvertes.fr/ujm-00721187
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Soumis le : jeudi 26 juillet 2012 - 17:47:20
Dernière modification le : vendredi 14 septembre 2018 - 09:16:05
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  • HAL Id : ujm-00721187, version 1
  • ARXIV : 1207.6463

Citation

François Lucas, Daniel Schaub, Mark Spivakovsky. On the Pierce-Birkhoff Conjecture. 2012. 〈ujm-00721187〉

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