The real field with convergent generalized power series

Abstract

Abstract. We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on [0, 1] by a series cnx αn with 0 ≤ αn → ∞ and |cn|r αn < ∞ for some r > 1 is definable. This expansion is polynomially bounded.

Untitled

“…Let A ⊆ I n . In accordance with [8], we call A a Λ-set if A ∈ Λ n ; if in addition A is a manifold, we call A a Λ-manifold. Similarly, A is a sub-Λ-set if there are m ≥ n and B ∈ Λ m such that A = Π n (B); if in addition A is a manifold, then A is a sub-Λ-manifold.…”

“…The idea for the proof of Theorem 1 is as follows. We try to follow the constructions of the o-minimal structures in [8,9]. The main ingredients there are a Weierstrass Preparation Theorem due to Tougeron and an adaptation of Gabrielov's fiber cutting argument to the non-Noetherian case.…”

“…We briefly present some key points of the Newton algorithm in Section 1, in such a way that our normalization algorithm, developed in Section 2, can readily be seen as a generalization of the two-variable case. In Sections 3 and 4 we combine this algorithm with the fiber cutting arguments of [8]. Finally, we prove Theorem 1 in Section 5.…”

10.4115/jla.v2i0.69

“…Then the green function restricted to the boundary is definable in the o-minimal structure R R an which is the expansion of R an by the power functions x → x λ for arbitrary real λ (see [9] and [27] for this o-minimal structure). Note that functions definable in R R an are exactly the functions with are piecewise given by convergent generalized power series as considered in [10] with support (i.e., the set of exponents, see [10, p. 4377]) contained in a finitely generated monoid over the nonnegative numbers. We also need "bad" approximation by rational numbers if the angle at a singular boundary point of the polygon is an irrational multiple of π in order to keep convergence when integrating globally subanalytic functions with respect to the harmonic measure.…”

“…But the globally subanalytic sets form an o-minimal structure denoted by R an . O-minimal structures provide an excellent framework to capture important concepts from analysis and different classes of functions in the tame setting (see also Van den Dries and Speissegger [10,11], Speissegger [37], Rolin et. al.…”