Short generating functions for some semigroup algebras
Summary: Let \(a_1,\dots,a_n\) be distinct, positive integers with \((a_1,...,a_n)=1\), and let k be an arbitrary field. Let \(H(a_1,...,a_n;z)\) denote the Hilbert series of the graded algebra k\([t^{a_1},t^{a_2},...,t^{a_n}]\). We show that, when \(n=3\), this rational function has a simple expression in terms of \(a_1,a_2,a_3\); in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any \(n \geq 4\).
scientific article; zbMATH DE number 1985377