A formal study of Bernstein coefficients and polynomials

Bernstein coe cients provide a discrete approximation of the behavior of a polynomial inside an interval. This can be used for example to isolate real roots of polynomials. We prove a criterion for the existence of a single root in an interval and the correctness of the de Casteljau algorithm to compute e ciently Bernstein coe cients. Key-words: Polynomials, de Casteljau, Bernstein polynomials, Coq, Constructive mathematics, Real root isolation ∗ This work has been partially funded by the Galapagos project, of the French ANR. † This work has been partially funded by the FORMATH project, nr. 243847, of the FET program within the 7th Framework program of the European Commission in ria -0 05 03 01 7, v er si on 2 22 S ep 2 01 0 A formal study of Bernstein coe cients and polynomials Résumé : Les coe cients de Bernstein fournissent une approximation discr`ete du comportement d'un polynôme sur un intervalle donné. Ils peuvent être par exemple utilisés pour isoler les racines réelles d'un polynôme. Nous prouvons un critère su sant à l'existence d'une unique racine dans un intervalle ainsi que la correction de l'algorithme de de Casteljau pour calculer e cacement les coe cients de Bernstein. Mots-clés : Polynômes, de Casteljau, polynômes de Bernstein, Coq, Mathématiques constructives, isolation de racines réelles in ria -0 05 03 01 7, v er si on 2 22 S ep 2 01 0 A formal study of Bernstein coe cients and polynomials 3