On the complexity of modular composition of generic polynomials

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. For generic g and a, 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 software development and comment on implementation results for the main building blocks in our composition algorithm.
Contains joint work with Seung Gyu Hyun, Bruno Salvy, Eric Schost, Gilles Villard.