Graded algebras with prescribed Hilbert series
Vesselin Drensky

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 with exponentially bounded nonnegative integer coefficients we suggest a simple construction of a finitely generated monomial associative algebra with Hilbert series very close to . If is rational/algebraic/transcendental, then the same is . If the growth of the coefficients of is polynomial, in the same way we construct a graded algebra preserving the polynomial growth of the coefficients of its Hilbert series . Applying a classical result of Fatou from 1906 we obtain that if a finitely generated graded algebra 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 with polynomial identity.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Advanced Algebra and Logic
