On Taylor polynomials of rational functions

par Simone Naldi

jeudi 4 déc. 2025, 11:30 → 12:30 Europe/Paris

XR203 (XLIM)

In this talk I will be interested in the following question: which polynomials F in C[x] in n variables, of degree m are Taylor polynomials of some rational function P/Q in C(x) with P of degree d and Q of degree e? This is related to the problem of determining Padé approximations of multivariate polynomials, a well known problem in computer algebra. In the univariate case, we will see that this leads to a natural determinantal condition on the coefficient of F (which was classically known). We will also see that this does not generalize directly to the multivariate case, but leads to open questions, concerning the defining equations of so-called Taylor varieties and their defectivity. If time allows, I will talk about possible applications (work in progress) to algebraic power series.