A generic effective Nullstellensatz (Q1774958)

J. Kollár has proved the following theorem, known as effective Nullstellensatz: Theorem. Let \(F\) be a field, and \(h_{1 },\dots,h_{M}\in F[x_{1},\dots,x_{n} ] \) with maximum total degree \(\delta\) such that \((h_{1 },\dots,h_{M})= F[x_{1},\dots,x_{n} ] \). Then there exists \(g_{1 },\dots,g_{M}\in F[x_{1},\dots,x_{n} ] \) with maximum total degree \(\delta^{ \min (n,M) } \) such that \(\sum_{i=1}^M g_{i} h_{i} =1\). The aim of this paper is to give a generic effective Nullstellensatz, that is assuming that \(h_{1 },\dots,h_{M}\) are generic. In this case the generic bound is small and is linear in the degrees of the polynomials. Also the set \(\{g_{1 },\dots,g_{M}\}\) is obtained by linear algebra.