Inventiones mathematicae

, Volume 162, Issue 1, pp 1–17

On the effective Nullstellensatz


DOI: 10.1007/s00222-004-0434-8

Cite this article as:
Jelonek, Z. Invent. math. (2005) 162: 1. doi:10.1007/s00222-004-0434-8


Let \(\mathbb{K}\) be an algebraically closed field and let \(X\subset\mathbb{K}^m\) be an n-dimensional affine variety. Assume that f1,...,fk are polynomials which have no common zeros on X. We estimate the degrees of polynomials \(A_i\in\mathbb{K}[X]\) such that 1=∑ki=1Aifi on X. Our estimate is sharp for kn and nearly sharp for k>n. Now assume that f1,...,fk are polynomials on X. Let \(I=(f_1,\dots,f_k)\subset\mathbb{K}[X]\) be the ideal generated by fi. It is well-known that there is a number e(I) (the Noether exponent) such that √Ie(I)I. We give a sharp estimate of e(I) in terms of n, deg X and deg fi. 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 \({\deg} f_jg_{ij}\le\prod^n_{i=1}\deg f_i\).

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Instytut MatematycznyPolska Akademia NaukKrakówPoland

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