Smooth parameterizations of power-subanalytic sets and compositions of Gevrey functions

Abstract

We show that if $X$ is an $m$ -dimensional definable set in $\mathbb {R}_\text {an}^\text{pow}$ , the structure of real subanalytic sets with real power maps added, then for any positive integer $r$ there exists a $C^{r}$ -parameterization of $X$ consisting of $cr^{m^{3}}$ maps for some constant $c$ . Moreover, these maps are real analytic and this bound is uniform for a definable family.

Mild parametrizations of power-subanalytic sets

“…The following parametrization theorem is the first main result of this paper. It further refines the main result of [VH21b], which made the polynomial dependence on r, obtained by Cluckers, Pila and Wilkie in [CPW20], more explicit. More precisely, we improve the degree of r from m 3 to m. This is the same amount of charts as in [BN19] for families of subanalytic sets.…”

“…Let us now give some elementary properties of mild functions. The second and third result can be found in [VH21b], the first one is clear.…”

“…We now state the Pre-parametrization Theorem that is the key result in the proofs of Theorem 4.1 and Theorem 4.6. It is the version from [VH21b], but it is originally due to Cluckers, Pila and Wilkie in [CPW20].…”

“…Mild functions were introduced by Pila in [Pil06] in order to bound the number of rational points on a Pfaffian curve. We will use the definition of [VH21b], where I derived a result on the composition of such functions by translating them to Gevrey functions (see [Gev18]), and using the results that are known for these functions.…”

“…They were mainly interested in the diophantine applications and obtain some similar results as in [BN19], but in the larger class of power-subananalytic sets. In view of the above, it seems natural to make this polynomial in r more explicit in terms of the combinatorial data defining X. I have shown in [VH21b] that the polynomial has degree m 3 by precisely analyzing their method. In this paper, I improve their construction such that the degree is just m (as in [BN19], but for the more general class of power-subanalytic sets), see Theorem 4.1.…”