Estimates for Weierstrass division in ultradifferentiable classes

TL;DR

This paper investigates the Weierstrass division theorem within ultradifferentiable classes, establishing bounds on the regularity of quotients and remainders, and providing examples that optimize these bounds, thus refining previous results.

Contribution

It introduces new estimates for Weierstrass division in Denjoy-Carleman classes, relating the regularity of solutions to the geometry of polynomial roots, and improves upon earlier bounds.

Findings

Quotients and remainders can be chosen in classes with controlled regularity

The ojasiewicz exponent  relates to root geometry and bounds regularity

Examples show the bounds are sharp and sometimes strictly less than the polynomial degree

Abstract

We study the Weierstrass division theorem for function germs in strongly non-quasianalytic Denjoy-Carleman classes $\mathcal{C}_M$. For suitable divisors $P(x,t)=x^d+a_1(t)x^{d-1}+\cdots+a_d(t)$ with real-analytic coefficients $a_j$, we show that the quotient and the remainder can be chosen of class $\mathcal{C}_{M^\sigma}$, where $M^\sigma=((M_j)^\sigma)_{j\geq 0}$ and $\sigma$ is a certain {\L}ojasiewicz exponent $\sigma$ related to the geometry of the roots of $P$ and verifying $1\leq \sigma\leq d$. We provide various examples for which $\sigma$ is optimal, in particular strictly less than $d$, which sharpens earlier results of Bronshtein and of Chaumat-Chollet.