Estimates for Weierstrass division in ultradifferentiable classes

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 $M^{\sigma }=((M_j{)}^{\sigma }{)}_{j\ge 0}$ and $\sigma$ is a certain {\L}ojasiewicz exponent $\sigma$ related to the geometry of the roots of $P$ and verifying $1\le \sigma \le d$. 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.