Approximate roots of a valuation and the Pierce-Birkhoff conjecture

F. Lucas; J. Madden; D. Schaub; M. Spivakovsky

doi:10.5802/afst.1336

F. Lucas ¹; J. Madden ²; D. Schaub ¹; M. Spivakovsky ³

¹ Département de Mathématiques/CNRS UMR 6093, Université d’Angers, 2, bd Lavoisier, 49045 Angers cédex, France
² Department of Mathematics, Louisiana State University at Baton Rouge, Baton Rouge, LA, USA
³ Inst. de Mathématiques de Toulouse/CNRS 5219, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cédex 9, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. 2, pp. 259-342.

Abstract
Abstract

In this paper, we construct an object, called a system of approximate roots of a valuation, centered in a regular local ring, which describes the fine structure of the valuation (namely, its valuation ideals and the graded algebra). We apply this construction to valuations associated to a point of the real spectrum of a regular local ring $A$ . We give two versions of the construction: the first, much simpler, in a special case (roughly speaking, that of rank 1 valuations), the second – in the case of complete regular local rings and valuations of arbitrary rank.

We then describe certain subsets $C \subset S p e r A$ by explicit formulae in terms of approximate roots; we conjecture that these sets satisfy the Connectedness (respectively, Definable Connectedness) conjecture. Establishing this for a certain regular ring $A$ would imply that $A$ is a Pierce-Birkhoff ring (this means that the Pierce-Birkhoff conjecture holds in $A$ ).

Finally, we use these constructions and results to prove the Definable Connectedness conjecture (and hence a fortiori the Pierce-Birkhoff conjecture) in the special case when $dim A = 2$ .

Les résultats contenus dans ce papier constituent une étape dans notre tentative de démontrer la Conjecture de Pierce-Bikhoff pour des anneaux réguliers en toute dimension (et en particulier la conjecture classique pour un anneau de polynômes sur un corps réel clos). On commence par rappeler les conjectures de Connexité et de Connexité Définissable, qui ont toutes deux pour conséquence la conjecture de Pierce-Birkhoff.

Nous introduisons alors la notion de système de racines approchées pour une valuation $ν$ sur un anneau $A$ : c’est une collection $Q$ d’éléments de $A$ telle que tout $ν$ -idéal est engendré par un produit d’éléments de $Q$ . On se sert alors des racines approchées pour définir, par des formules explicites, des sous-ensembles du spectre réel de $A$ , fortement susceptibles de vérifier la conjecture de Connexité Définissable.

On prouve ainsi la conjecture de Pierce-Birkhoff pour un anneau régulier arbitraire de dimension 2.

MR Zbl

DOI: 10.5802/afst.1336

Author's affiliations:

F. Lucas ¹; J. Madden ²; D. Schaub ¹; M. Spivakovsky ³

¹ Département de Mathématiques/CNRS UMR 6093, Université d’Angers, 2, bd Lavoisier, 49045 Angers cédex, France
² Department of Mathematics, Louisiana State University at Baton Rouge, Baton Rouge, LA, USA
³ Inst. de Mathématiques de Toulouse/CNRS 5219, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cédex 9, France

@article{AFST_2012_6_21_2_259_0,
     author = {F. Lucas and J. Madden and D. Schaub and M. Spivakovsky},
     title = {Approximate roots of a valuation and the {Pierce-Birkhoff} conjecture},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {259--342},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 21},
     number = {2},
     year = {2012},
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     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1336/}
}

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F. Lucas; J. Madden; D. Schaub; M. Spivakovsky. Approximate roots of a valuation and the Pierce-Birkhoff conjecture. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. 2, pp. 259-342. doi : 10.5802/afst.1336. https://afst.centre-mersenne.org/articles/10.5802/afst.1336/

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