BOUNDS FOR THE HILBERT-FUNCTION OF POLYNOMIAL IDEALS AND FOR THE DEGREES IN THE NULLSTELLENSATZ

We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees o f the input polynomials but also on an additional parameter called the geometric degree of the system of equations. The obtained bound is po lynomial in these parameters. It is essentially optimal in the general case, and it substantially improves the existent bounds in some speci al cases. The proof of this result is combinatorial, and relies on glo bal estimates for the Hilbert function of homogeneous polynomial ideal s. 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 r adical polynomial ideal. (C) 1997 Elsevier Science B.V.