Let $R$ be a positively graded algebra over field. We say that is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has all roots on unit circle. Such rings arise naturally in commutative algebra, numerical semigroup theory and Ehrhart theory. If standard graded, we prove that, under additional hypothesis Koszul or an irreducible $h$-polynomial, algebras coincide with complete i...

Toric ideals to hierarchical models are invariant under the action of a product symmetric groups. Taking number factors, say m, into account, we introduce and study filtrations their equivariant Hilbert series. We present condition that guarantees series is rational function in m+1 variables with coefficients. Furthermore give explicit formulas for functions coefficients field an algorithm dete...

Let $$\mathcal {P}$$ be a simple thin polyomino, namely polyomino that has no holes and does not contain square tetromino as subpolyomino. In this paper, we determine the reduced Hilbert–Poincaré series $$h(t)/(1-t)^d$$ of $$K[\mathcal {P}]$$ by proving h(t) is rook polynomial . As an application, characterize Gorenstein polyominoes.

Motivating the perturbations of frames in Hilbert and Banach spaces, in this paper we introduce the invariance of Fr'echet frames under perturbation. Also we show that for any Fr'echet spaces, there is a Fr'echet frame and any element in these spaces  has a series expansion.

We consider graded finitely presented algebras and modules over a field. Under some restrictions, the set of Hilbert series of such algebras (or modules) becomes finite. Claims of that types imply rationality of Hilbert and Poincare series of some algebras and modules, including periodicity of Hilbert functions of common (e.g., Noetherian) modules and algebras of linear growth.

It is not possible to determine from the Hilbert series whether a graded Noetherian algebra is a complete intersection. Nevertheless, such Hilbert series satisfy very stringent conditions and we define a concept CI-type for formal power series, that embodies some of these necessary properties. This definition works for algebras that are not standard i. e. not generated in degree 1. For the clas...