Hilbert series of noncommutative algebras and context-free languages
The Hilbert series is one of the most important algebraic invariants of innite-dimensional
graded associative algebra. The noncommutative Groebner basis machine reduces the prob-
lem of nding Hilbert series to the case of monomial algebra.
We apply both noncommutative and commutative Groebner bases theory as well as the
theory of formal languages to provide a new method for symbolic computation of Hilbert
series of graded associative algebras. Whereas in general the problem of computation oh
such a Hilbert series is known to be algorithmically unsolvable, we have describe a general
class of algebras (called homologically unambiguous) with unambiguous context-free set of
relations for which our method give eective algorithms. Unlike previously known methods,
our algorithm is applicable to algebras with irrational Hilbert series and produces an alge-
braic equation which denes the series. The examples include innitely presented monomials
algebras as well as nitely presented algebras with irrational Hilbert series such that the
associated monomial algebras are homologically unambiguous.