Graded algebras with prescribed Hilbert series

TL;DR

This paper presents a method to construct finitely generated monomial algebras with Hilbert series closely matching any given power series with nonnegative integer coefficients, revealing a dichotomy in their possible series types based on growth properties.

Contribution

It introduces a simple construction technique for algebras with prescribed Hilbert series and establishes a dichotomy for series types based on Gelfand-Kirillov dimension.

Findings

Constructs algebras with Hilbert series approximating any given power series.

Shows Hilbert series are either rational or transcendental for algebras with finite Gelfand-Kirillov dimension.

Establishes the same dichotomy for algebras with polynomial identities.

Abstract

For any power series $a(t)$ with exponentially bounded nonnegative integer coefficients we suggest a simple construction of a finitely generated monomial associative algebra $R$ with Hilbert series $H(R,t)$ very close to $a(t)$. If $a(t)$ is rational/algebraic/transcendental, then the same is $H(R,t)$. If the growth of the coefficients of $a(t)$ is polynomial, in the same way we construct a graded algebra $R$ preserving the polynomial growth of the coefficients of its Hilbert series $H(R,t)$. Applying a classical result of Fatou from 1906 we obtain that if a finitely generated graded algebra $R$ has a finite Gelfand-Kirillov dimension, then its Hilbert series is either rational or transcendental. In particular the same dichotomy holds for the Hilbert series of finitely generated algebras $R$ with polynomial identity.