M/E23

The topic of the talk evolves around the following question: which polynomials of degree $m$ in $n$ variables are the Taylor polynomials of a rational function of degree $(d,e)$? In other words, what conditions must satisfy the coefficients of a polynomial to let it be the Taylor expansion of a rational function? For every choice of the parameters $(n,d,e,m)$ this defines a projective variety in the set of degree m polynomials, that we call the Taylor variety. In the talk I will present some results concerning the dimension theory and remarkable properties of Taylor varieties. The main motivation comes from numerical analysis, precisely, from the so-called Padé approximation problem, where the polynomial is the Taylor polynomial of a regular function around the origin, and one looks for a rational function of low degree that has the same Taylor polynomial. Joint work with A. Conca, G. Ottaviani and B. Sturmfels.

Simone Naldi

Limoges