This talk is about algorithms for modular composition of univariate
polynomials, and for computing minimal polynomials. For two univariate
polynomials a and g over a commutative field, modular composition asks to
compute h(a) mod g for some given h, while the minimal polynomial problem is
to compute h of minimal degree such that h(a) = 0 mod g. We propose algorithms
whose complexity bound improves upon previous algorithms and in particular upon
Brent and Kung's approach (1978); the new complexity bound is subquadratic in
the degree of g and a even when using cubic-time matrix multiplication. Our
improvement comes from the fast computation of specific bases of bivariate
ideals,
and from efficient operations with these bases thanks to fast univariate
polynomial
matrix algorithms. We will also report on experimental results using the
Polynomial Matrix Library.

Contains joint work with Seung Gyu Hyun, Bruno Salvy, Eric Schost, Gilles
Villard.