Oscillating properties in the integration of \(x^\lambda\)-functions
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.
scientific article; zbMATH DE number 1686432