Displaying similar documents to “On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields”

Approximate roots of a valuation and the Pierce-Birkhoff conjecture

F. Lucas, J. Madden, D. Schaub, M. Spivakovsky (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

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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 –...

Separating ideals in dimension 2.

James J. Madden, Niels Schwartz (1997)

Revista Matemática de la Universidad Complutense de Madrid

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Experience shows that in geometric situations the separating ideal associated with two orderings of a ring measures the degree of tangency of the corresponding ultrafilters of semialgebraic sets. A related notion of separating ideals is introduced for pairs of valuations of a ring. The comparison of both types of separating ideals helps to understand how a point on a surface is approached by different half-branches of curves.