Related papers: Computation of Poincare-Betti series for monomial …

Let R be the quotient of a polynomial ring over a field k by an ideal generated by monomials. We derive a formula for the multigraded Poincare' series of R, i.e., the generating function for the ranks of the modules in a minimal multigraded…

Let $S=\Bbbk[x_1,..., x_n]$ be a polynomial ring over a field $\Bbbk$ and $I$ a monomial ideal of $S$. It is well known that the Poincar\'e series of $\Bbbk$ over $S/I$ is rational. We describe the coefficients of the denominator of the…

We give a comparison between the Poincare series of two monomial rings: $R=A/I$ and $R_q=A/I_q$ where $I_q$ is a monomial ideal generated by the $q$'th power of monomial generators of $I$. We compute the Poincare series for a class of…

Let $S=\mathbb{K}[x_1,\ldots,x_n]$ the polynomial ring over a field $\mathbb{K}$. In this paper for some families of monomial ideals $I \subset S$ we study the minimal number of generators of $I^k$. We use this results to find some other…

In this paper, we present several algorithms for dealing with graded components of Laurent polynomial rings. To be more precise, let $S$ be the Laurent polynomial ring $k[x_1,...,x_{r},x_{r+1}^{\pm 1},..., x_n^{\pm 1}]$, $k$ algebraicaly…

A formula for computation of the bivariate Poincar\'e series $\mathcal{P}_d(z,t)$ for the algebra of covariants of binary $d$-form is found.

This paper gives a sharp upper bound for the Betti numbers of a finitely generated multigraded $R$-module, where $R=\Bbbk [x_{1},...,x_{m}]$ is the polynomial ring over a field $\Bbbk$ in $m$ variables. The bound is given in terms of the…

For an affine toric variety X we compute the Poincare series of the multi-index filtration defined by a finite number of monomial divisorial valuations on the ring O_{X,0}. We give an alternative description of the Poincare series as an…

This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its…

We investigate polynomial sets {P n } n$\ge$0 with generating power series of the form F (xt -- R(t)) and satisfying, for n $\ge$ 0, the (d + 1)-order recursion xP\_ n (x) = P\_{ n+1 }(x) +\sum\_{ l=0}^{d} \gamma^{l}\_{n} P\_{ n--l} (x),…

The Concepts of poly-Bernoulli numbers $B_n^{(k)}$, poly-Bernoulli polynomials $B_n^{k}{(t)}$ and the generalized poly-bernoulli numbers $B_{n}^{(k)}(a,b)$ are generalized to $B_{n}^{(k)}(t,a,b,c)$ which is called the generalized…

We offer a Maple package {\tt Poincare\_Series} for calculating the Poincar\'e series for the algebras of invariants/covariants of binary forms, for the algebras of joint invariants/covariants of several binary forms, for the kernel of…

Let $k$ be a field and $x,y$ indeterminates over $k$. Let $R=k[x^a,x^{p_1}y^{s_1},\ldots,x^{p_t}y^{s_t},y^b] \subseteq k[x,y]$. We calculate the Hilbert polynomial of $(x^a,y^b)$. The multiplicity of this ideal provides part of a criterion…

The bigraded Betti numbers b^{-i,2j}(P) of a simple polytope P are the dimensions of the bigraded components of the Tor groups of the face ring k[P]. The numbers b^{-i,2j}(P) reflect the combinatorial structure of P as well as the topology…

A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^{\alpha}\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number…

Let $R=\mathbb{K}[x_1,\dots,x_n]$, a graded algebra $S=R/I$ satisfies $N_{k,p}$ if $I$ is generated in degree $k$, and the graded minimal resolution is linear the first $p$ steps, and the $k$-index of $S$ is the largest $p$ such that $S$…

Hyperquot schemes are generalizations of Grothendieck's Quot scheme to partial flags. Using a Bialynicki-Birula decomposition, we obtain combinatorial data for the Betti numbers, and collect this information into the form of rational…

A hypergraph $H=(V,E)$, where $V=\{x_1,...,x_n\}$ and $E\subseteq 2^V$ defines a hypergraph algebra $R_H=k[x_1,...,x_n]/(x_{i_1}... x_{i_k}; \{i_1,...,i_k\}\in E)$. All our hypergraphs are $d$-uniform, i.e., $|e_i|=d$ for all $e_i\in E$. We…

The aim of this paper is to introduce a method for computing Hilbert decompositions (and consequently the Hilbert depth) of a finitely generated multigraded module $M$ over the polynomial ring $K[X_1,..., X_n]$ by reducing the problem to…

In this paper we study standard bases for submodules of a mixed power series and polynomial ring $R[[t_1,\ldots,t_m]][x_1,\ldots,x_n]^s$ respectively of their localization with respect to a $t$-local monomial ordering for a certain class of…