We present a new linearization method to over-approximate non-linear multivariate polynomials with convex polyhedra. It is based on Handelman-Krivine's theorem and consists in using products of constraints of a polyhedron to over-approximate a polynomial on this polyhedron. We implemented it together with two other linearization methods that we will not detail in this paper, but that we shall use as comparison. Our implementation in Ocaml generates certificates that can be verified by a trusted checker, certified in Coq, that guarantees the correctness of our linear approximation.