Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz

TL;DR

This paper introduces a new effective Nullstellensatz with degree bounds depending on the geometric degree of the system, providing polynomial bounds that are optimal in general and improve existing bounds in special cases.

Contribution

It establishes a novel degree bound for the Nullstellensatz that incorporates the geometric degree, using combinatorial methods and Hilbert function estimations.

Findings

Degree bounds are polynomial in system parameters.

Bounds are essentially optimal in the general case.

Improves upon existing bounds in certain special cases.

Abstract

We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the {\it geometric degree of the system of equations}. The obtained bound is polynomial in these parameters. It is essentially optimal in the general case, and it substantially improves the existent bounds in some special cases. The proof of this result is combinatorial, and it relies on global estimations for the Hilbert function of homogeneous polynomial ideals. In this direction, we obtain a lower bound for the Hilbert function of an arbitrary homogeneous polynomial ideal, and an upper bound for the Hilbert function of a generic hypersurface section of an unmixed radical polynomial ideal.