Bigatti@UniMat.To.CNR.It Let R := k[X 1 , . . . , X N ] be the polynomial ring in N indeterminates over a field k of characteristic 0 with deg(X i ) = 1 for i = 1, . . . , N , and let I be a homogeneous ideal of R . The Hilbert function of I is the function from N to N which associates to every natural number d the dimension of I d as a k -vectorspace.I has an essentially unique minimal graded free resolutionwhich is characterized, among the free graded resolutions, by the conditionAnd therefore the Betti numbers, which are defined by β q (I) := rankL q

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Abstract.F. S. Macaulay gave necessary and sufficient conditions on the growth of a nonnegative integer-valued function which determine when such a function can be the Hubert function of a standard graded A:-algebra. We investigate some algebraic and geometric consequences which arise from the extremal cases of Macaulay's theorem. Our work also builds on the fundamental work of G. Gotzmann.Our principal applications are to the study of Hubert functions of zeroschemes with uniformity conditions. As a consequence, we have new strong limitations on the possible Hubert functions of the points which arise as a general hyperplane section of an irreducible curve.

We address the problem of computing ideals of polynomials which vanish at a finite set of points. In particular we develop a modular Buchberger-Moeller algorithm, best suited for the computation over QQ, and study its complexity; then we describe a variant for the computation of ideals of projective points, which uses a direct approach and a new stopping criterion. The described algorithms are implemented in cocoa, and we report some experimental timings

Toric ideals are binomial ideals which represent the algebraic relations of sets of power products. They appear in many problems arising from different branches of mathematics. In this paper, we develop new theories which allow us to devise a parallel algorithm and an efficient elimination algorithm. In many respects they improve existing algorithms for the computation of toric ideals