Estimates for Weierstrass division in ultradifferentiable classes

We study the Weierstrass division theorem for function germs in strongly non-quasianalytic Denjoy–Carleman classes CMCM. For suitable divisors P(x,t)=xd+a1(t)xd−1+⋯+ad(t)P(x,t)=xd+a1(t)xd−1+⋯+ad(t) with real-analytic coefficients ajaj, we show that the quotient and the remainder can be chosen of class CMσCMσ, where Mσ=((Mj)σ)j≥0Mσ=((Mj)σ)j≥0 and σ is a certain Łojasiewicz exponent related to the geometry of the roots of P   and verifying 1≤σ≤d1≤σ≤d. We provide various examples for which σ is optimal, in particular strictly less than d, which sharpens earlier results of Bronshtein and of Chaumat–Chollet.