Lexsegment Ideals and Their h-Polynomials

Abstract

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