Abstract
Hilbert's Nullstellensatz is a cornerstone in Algebraic Geometry. It describes by a Bézout identity when polynomial equations do not share any root in an algebraically closed field. Here I will present sharp results on the heights and degrees of polynomials showing up in the Bezout identity, when the polynomials defining field admits a height notion. And hopefully I will be able to comment on some applications of these results to complexity problems over finite fields.