Nullstellensatz effectif et Conjecture de Serre (Théorème de Quillen‐Suslin) pour le Calcul Formel

Abstract

Abstract Let k be an arbitrary field, X 1 ,…., X n indeterminates over k and F 1 …, F 3 ε ∈ k[X 1 …,X n ] polynomials of maximal degree \documentclass{article}\pagestyle{empty}\begin{document}$ d: = \mathop {\max }\limits_{1 \le i \le a} \deg $\end{document} ( F i ). We give an elementary proof of the following effective Nullstellensatz: Assume that F 1 ,…, F have no common zero in the algebraic closure of k. Then there exist polynomials P 1 …, P 3 ε ∈ k[X 1 …,X n ] such that \documentclass{article}\pagestyle{empty}\begin{document}$ 1: = \mathop \Sigma \limits_{1 \le i \le a} $\end{document} P i F i and This result has many applications in Computer Algebra. To exemplify this, we give an effective quantitative and algorithmic version of the Quillen‐Suslin Theorem baaed on our effective Nullstellensatz.