Oscillating properties in the integration of \(x^\lambda\)-functions (Q2761874)

J. M. Lion and J. P. Rolin studied the integrals of global subanalytic functions on global subanalytic sets and proved that such integrals belong to the class of logarithmico-exponential (LE) functions.NEWLINENEWLINE ``Global'' means working in the projective space and replaces advantageously ``relatively compact'' of Łojasiewicz and his group. Gabrielov showed that global subanalytic functions form an \(o\)-minimal structure.NEWLINENEWLINE The author deals with a larger \(o\)-minimal structure of \(x^\lambda\)-functions, introduced by Miller and Tougeron. These are finite compositions of global subanalytic functions and power maps (the latter prolongated by 0 where necessary).NEWLINENEWLINE A power map writes \(\Gamma (x_1,\dots, x_p)=(x_1^{\gamma_1},\dots,x_p^{\gamma_p})\). Thus \(\gamma=(\gamma_1, \dots, \gamma_p)\) is a parameter. A function \(h_k\circ\Gamma_k\circ\cdots\circ h_1\circ \Gamma_1\), with \(h_i\) global subanalytic and \(\Gamma_i\) power maps will be denoted \(f_\gamma\), where \(\gamma=(\gamma_1,\dots,\gamma_p)\) all the powers that appear in \(\Gamma_i\), \(i=1,\dots,k\).NEWLINENEWLINE With this notation, the author proves Theorem 1. If \(f_\gamma:\mathbb{R}^{n+1}\to \mathbb{R}\) is an \(x^\lambda\)-function with \(\gamma \in\mathbb{R}^N\), then there exists a subset \(D_\gamma\subset\mathbb{R}^N\) of full Lebesgue measure such that for all \(\gamma\in D_\gamma\) and all continuous \(x^\lambda\)-functions \(\varphi<\psi\) defined on \(\mathbb{R}^n\) the integral NEWLINE\[NEWLINEF_\gamma (x)= \int^{\psi(x)}_{\varphi(x)} f_\gamma(x,y)dy\text{ is again an } x^\lambda\text{-function on }\mathbb{R}^n.NEWLINE\]NEWLINE We have thus the good non-oscillation properties of the integrals of almost all \(x^\lambda\)-functions, but certainly not all (an idea of a simple counterexample is given).NEWLINENEWLINE Only an idea of the proof is given, as this is a short note of CRAS. The proof is based on Lion-Rolin preparation theorem for \(x^\lambda\)-functions, which permits to integrate them formally and see that an \(x^\lambda\)-function results from this integration only if some diophantine equations on \(\gamma_i\) are verified. It remains to see that the set of good \(\gamma\) is of full measure. To show that, metric properties of real algebraic sets (from Comté-Yomdin) are used.NEWLINENEWLINE It would be most interesting to see the proof published and answer the question when the set \(D_f\) is described in an explicit way.NEWLINENEWLINE Theorem 2 shows that for almost all parameters the number of connected components of the level sets of integrals \(F\) is uniformly bounded.NEWLINENEWLINE This is a natural consequence of non-oscillation.