  For a given computational problem, a certificate is a piece of data that one
(the prover) attaches to the output with the aim of allowing efficient
verification (by the verifier) that this output is correct. Here, we consider
the minimal approximant basis problem, for which the fastest known algorithms
output a polynomial matrix of dimensions $m \times m$ and average degree $D/m$
using $O\tilde{~}(m^\omega \frac{D}{m})$ field operations. We propose a
certificate which, for typical instances of the problem, is computed by the
prover using $O(m^\omega \frac{D}{m})$ additional field operations and allows
verification of the approximant basis by a Monte Carlo algorithm with cost
bound $O(m^\omega + m D)$.
  Besides theoretical interest, our motivation also comes from the fact that
approximant bases arise in most of the fastest known algorithms for linear
algebra over the univariate polynomials; thus, this work may help in designing
certificates for other polynomial matrix computations. Furthermore,
cryptographic challenges such as breaking records for discrete logarithm
computations or for integer factorization rely in particular on computing
minimal approximant bases for large instances: certificates can then be used to
provide reliable computation on outsourced and error-prone clusters.
