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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 Sér. 6, 21 no. 2 (2012), p. 259-342, doi: 10.5802/afst.1336
Article PDF | Reviews MR 2978097 | Zbl 1271.13051

Résumé - 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 $\dim \ A=2$.

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