Approximate Roots of a Valuation and the Pierce–birkhoff Conjecture

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 ⊂ 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.