Hilbert's Nullstellensatz - Effective Nullstellensatz

Effective Nullstellensatz

In all of its variants, Hilbert's Nullstellensatz asserts that some polynomial belongs or not to an ideal generated, say, by ; we have in the strong version, in the weak form. This means the existence or the non existence of polynomials such that The usual proofs of the Nullstellensatz are non effective in the sense that they do not give any way to compute the .

This is thus a rather natural question to ask if there an effective way to compute the (and the exponent in the strong form) or to prove that they do not exist. To solve this problem, it suffices to provide an upper bound on the total degree of the : such a bound reduces the problem to a finite system of linear equations that may be solved by usual linear algebra techniques. Any such upper bound is called an effective Nullstellensatz.

A related problem is the ideal membership problem, which consists in testing if a polynomial belongs to an ideal. For this problem also, a solution is provided by a upper bound on the degree of the A general solution of the ideal membership problem provides an effective Nullstellensatz, at least for the weak form.

In 1925, Grete Hermann gave a upper bound for ideal membership problem that is doubly exponential in the number of variables. In 1982 Mayr and Meyer gave an example where the have a degree which is at least double exponential, showing that every general upper bound for the ideal membership problem is doubly exponential in the number of variables.

Until 1987, nobody had the idea that effective Nullstellensatz was easier than ideal membership, when Brownawell gave an upperbound for the effective Nullstellensatz which is simply exponential in the number of variables. Brownawell proof uses calculus techniques and thus is valid only in characteristic 0. Soon after, in 1988, János Kollár gave a purely algebraic proof valid in any characteristic, leading to a better bound.

In the case of the weak Nullstellensatz, Kollár's bound is the following:

Let be polynomials in n≥2 variables, of total degree If there exist polynomials such that then they can be chosen such that This bound is optimal if all the degrees are greater than 2.

If d is the maximum of the degrees of the, this bound may be simplified to

Kollár's result has been improved by several authors. M. Sombra has provided the best improvement, up to date, giving the bound . His bound is better than Kollár's as soon as at least two of the degrees that are involved are lower than 3..

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