Hilbert Series of a Quotient Algebra of Polynomials Original Research Article

Let R congruent with k[x1,..., xr]/(F1,..., Fn) where (F1,..., Fn) denotes the ideal of homogeneous polynomials F1,..., Fn of degree dk = deg Fk. Let us graduate R = R0 circled plus ··· circled plus Ri circled plus … by setting deg x1 = ··· = deg xr = 1 and define the Hilbert series of R by Hilb Rt = ∑i set membership, variant Ndim Riti. Then we have a lower bound (coefficientwise 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. Algebra15 (1987), 227-239] tells us that it holds for the first non-trivial degree, i.e., in degree 1 + min1 ≤ k ≤ ndk. In this paper we enlarge this result to a "wide" range of degrees.