O-minimal Preparation Theorems

Macintyre, Marker and Van den Dries [6] use model theory to show that globally subanalytic functions are piecewise given by terms in a certain language. Adding symbols for the exponential and logarithm function to this language, they also show that functions definable from exponentiation and globally subanalytic functions are piecewise given by terms. This piecewise definability by terms inspired Lion and Rolin [9] to find geometric proofs for what they call preparation theorems, which are sometimes more useful in applications. (A prepared function of several variables depends in a piecewise simple way on any chosen variable. An earlier preparation theorem for subanalytic functions is due to Parusinski [11]. Still earlier, Denef [2],[3] obtained results of this kind in the p-adic algebraic setting; see also Cluckers [1] for a recent p-adic analytic version and a nice application.) In Section 2 we prove a preparation theorem for functions belonging to a polynomially bounded o-minimal structure. In Section 3 we extend this to functions that are logarithmic-exponential over a polynomially bounded o-minimal structure. For a survey of o-minimality emphasizing geometry we refer to [5]. The rest of the introduction updates aspects of this survey and fixes notation and terminology. Recall from [5] that a structure on R is a family S = (Sn)n∈N such that for each n: (S1) Sn is a boolean algebra of subsets of R with R ∈ Sn, (S2) {(x, y) ∈ R : x = y} ∈ S2, (S3) the graphs of addition and multiplication on R belong to S3, (S4) A ∈ Sn implies A× R ∈ Sn+1 and R× A ∈ Sn+1,