arXiv subjects
Publications and source records attributed to Siegfried Van Hille.
The symmetric function theorem states that a polynomial that is invariant under permutation of variables, is a polynomial in the elementary symmetric polynomials. We deduce this classical result, in the analytic setting, from the multivariate Faà di Bruno formula. In two variables, this allows us to completely determine all coefficients that occur in the inductive equations.
We obtain two uniform parametrization theorems for families of bounded sets definable in $\mathbb R_{an}^{\mathbb R}$. Let $X = \{X_t \subset (0,1)^n \mid t \in T\}$ be a definable family of sets $X_t$ of dimension at most $m$. Firstly, $X_t$ admits a $C^r$-parametrization consisting of $cr^m$ maps for some positive constant $c = c(X)$, which is uniform in $t$. Secondly, $X_t$ admits a $C$-mild parametrization for any $C>1$, which is also uniform in $t$.
We show that if $X$ is an $m$-dimensional definable set in $\mathbb{R}^\text{pow}_\text{an}$, 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.
We prove that the function $P_α(x) = \exp(1-x^{-α})$ with $α> 0$, is $1/α$-mild. We apply this result to obtain a uniform $1/α$-mild parametrization of the family of curves $\{xy = ε^2 \mid (x,y) \in (0,1)^2\}$ for $ε\in (0,1)$, which does not have a uniform $0$-mild parametrization by work of Yomdin. More generally we can parametrize families of power-subanalytic curves. This improves a result of Benjamini and Novikov that gives a $2$-mild parametrization.