Symbolic Computation Group

The central objects of this talk are univariate polynomials over finite fields. We first review a methodology from analytic combinatorics that allows not only the study of the decomposition of polynomials into its irreducible factors but also the derivation of algorithmic properties as well as the estimation of average-case analysis of algorithms. This methodology can be naturally used to provide precise information on the factorization of polynomials into its irreducible factors similar to the classical problem of the decomposition of integers into primes. Examples of these results are provided. The shape of a random univariate polynomial over a finite field is also given.

Then, we briefly show several results for random polynomials over finite fields that were obtained using other methodologies based for example on probability and probabilistic combinatorics. We conclude providing several open problems of polynomials over finite fields related to number theory.