Lexsegment Ideals and Their h-Polynomials

Abstract

Let S = K[x1, . . . , xn] denote the polynomial ring in n variables over a field K with each deg xi = 1 and I ⊂ S a homogeneous ideal of S with dim S/I = d. The Hilbert series of S/I is of the form hS/I(λ)/(1 − λ)d, where hS/I(λ) = h0 + h1λ + h2λ2 + ⋯ + hsλs with hs≠ 0 is the h-polynomial of S/I. Given arbitrary integers r ≥ 1 and s ≥ 1, a lexsegment ideal I of S = K[x1,…,xn], where n ≤ max{r,s} + 2, satisfying reg(S/I) = r and deg hS/I(λ) = s will be constructed.