On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields

Sven Wagner

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Approximate roots of a valuation and the Pierce-Birkhoff conjecture

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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 –...

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