On the effective Nullstellensatz

Abstract

Let $\mathbb{K}$ be an algebraically closed field and let $X\subset\mathbb{K}^m$ be an n-dimensional affine variety. Assume that f₁,...,f _k are polynomials which have no common zeros on X. We estimate the degrees of polynomials $A_i\in\mathbb{K}[X]$ such that 1=∑^k_i=1A _i f _i on X. Our estimate is sharp for k≤n and nearly sharp for k>n. Now assume that f₁,...,f _k are polynomials on X. Let $I=(f_1,\dots,f_k)\subset\mathbb{K}[X]$ be the ideal generated by f _i . It is well-known that there is a number e(I) (the Noether exponent) such that √I^e(I)⊂I. We give a sharp estimate of e(I) in terms of n, deg X and deg f _i . We also give similar estimates in the projective case. Finally we obtain a result from the elimination theory: if $f_1,\dots,f_n\in\mathbb{K}[x_1,\dots,x_n]$ is a system of polynomials with a finite number of common zeros, then we have the following optimal elimination:

$$\phi_i(x_i)=\sum^n_{j=1}f_jg_{ij},\quad\ i=1,\dots,n,$$

where ${\deg} f_jg_{ij}\le\prod^n_{i=1}\deg f_i$.