HILBERT SERIES OF A QUOTIENT ALGEBRA OF P OLYNOMIALS

Let R congruent to k[x(1),..., x(r)]/(F-1,., F-n) where (F-1,..., F-n) denotes the ideal of homogeneous polynomials F-1,..., F-n of degree d (k) = deg F-k. Let us graduate R = R(0) + ... R(i) +... by setting deg x(1) deg x(r) = 1 and define the Hilbert series of R by Hilb R(t) = S igma(i is an element of N)dim R(i)t(i). Then we have a lower bound (co effi cientwise order). an old conjecture says that this lower bound is ''generically'' attained. The only general result (any r, any n) due to [M. Hochster and D. Laksov, Comm. Algebra 15 (1987), 227-239] tells us that it holds for the first non-trivial degree, i.e., in degree 1 + min(1 less than or equal to k less than or equal to n)d(k). In this paper we enlarge this result to a ''wide'' range of degrees. (C) 1995 Academic Press, Inc.