On the effective Nullstellensatz

Let K be an algebraically closed field and let X subset of K-m be an n-dimensional affine variety. Assume that f(1),..., f(k) are polynomials which have no common zeros on X. We estimate the degrees of polynomials A(i) is an element of K[X] such that 1 = Sigma(k)(i=1) A(i) f(i) n X. Our estimate is sharp for k <= n and nearly sharp for k > n. Now assume that f(1),..., f(k) are polynomials on X. Let I = (f(1),..., f(k)). K[X] be the ideal generated by fi. It is well-known that there is a number e(I) (the Noether exponent) such that root I-e(I) subset of 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), ..., f(n) is an element of K[x(1),..., 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)) = Sigma(n)(j=1) f(j)g(ij), i = 1,..., n, where deg f(j)g(ij) <= Pi(n)(i=1) deg f(i).