Approximate roots of a valuation and the Pierce-Birkhoff conjecture

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 (2012)

  • Volume: 21, Issue: 2, page 259-342
  • ISSN: 0240-2963

Abstract

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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 ⊂ 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 .

How to cite

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Lucas, F., et al. "Approximate roots of a valuation and the Pierce-Birkhoff conjecture." Annales de la faculté des sciences de Toulouse Mathématiques 21.2 (2012): 259-342. <http://eudml.org/doc/251011>.

@article{Lucas2012,
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 $C\subset \mbox \{Sper\}\ A$ 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 $\dim \ A=2$.},
affiliation = {Département de Mathématiques/CNRS UMR 6093, Université d’Angers, 2, bd Lavoisier, 49045 Angers cédex, France; Department of Mathematics, Louisiana State University at Baton Rouge, Baton Rouge, LA, USA; Département de Mathématiques/CNRS UMR 6093, Université d’Angers, 2, bd Lavoisier, 49045 Angers cédex, France; Inst. de Mathématiques de Toulouse/CNRS 5219, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cédex 9, France},
author = {Lucas, F., Madden, J., Schaub, D., Spivakovsky, M.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {valuation; regular local ring; approximate root; Pierce-Birkhoff conjecture; connectedness conjecture},
language = {eng},
month = {4},
number = {2},
pages = {259-342},
publisher = {Université Paul Sabatier, Toulouse},
title = {Approximate roots of a valuation and the Pierce-Birkhoff conjecture},
url = {http://eudml.org/doc/251011},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Lucas, F.
AU - Madden, J.
AU - Schaub, D.
AU - Spivakovsky, M.
TI - Approximate roots of a valuation and the Pierce-Birkhoff conjecture
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2012/4//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - 2
SP - 259
EP - 342
AB - 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 $C\subset \mbox {Sper}\ A$ 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 $\dim \ A=2$.
LA - eng
KW - valuation; regular local ring; approximate root; Pierce-Birkhoff conjecture; connectedness conjecture
UR - http://eudml.org/doc/251011
ER -

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