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

Let $R$ be a commutative ring, $f \in R[X_1,\ldots,X_k]$ a multivariate polynomial, and $G$ a finite subgroup of the group of units of $R$ satisfying a certain constraint, which always holds if $R$ is a field. Then, we evaluate $\sum…

We consider the mixed states of the bipartite quantum system with the first party a qubit and the second a qutrit. The group of local unitary transformations of the system, ignoring the overall phase factor, is the direct product G of SU(2)…

Let M be the set of mixed states and S the set of separable states of the two-qubit system, and G = SU(2) x SU(2) the group of local unitary transformations (ignoring the overall phase factor). We compute the multigraded Poincare series for…

Given any fixed integer $q\ge 2$, a $q$-monomial is of the format $\displaystyle x^{s_1}_{i_1}x^{s_2}_{i_2}...x_{i_t}^{s_t}$ such that $1\le s_j \le q-1$, $1\le j \le t$. $q$-monomials are natural generalizations of multilinear monomials.…

Let S be a polynomial ring in n variables, over an arbitrary field. We give the total, graded, and multigraded Betti numbers of S/M, for every monomial ideal M in S. We also give an explicit characterization of all monomial ideals M in S…

We introduce to the context of multigraded modules the methods of modules over categories from algebraic topology and homotopy theory. We develop the basic theory quite generally, with a view toward future applications to a wide class of…

Let $\mathcal{R} = \mathbb{K}[x_1, \dots, x_n]$ be a multivariate polynomial ring over a field $\mathbb{K}$ of characteristic 0. Consider $n$ algebraically independent elements $g_1, \dots, g_n$ in $\mathcal{R}$. Let $\mathcal{S}$ denote…

The Poincar\'e polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space $G/B$, while the $h$-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety.…

The purpose of this note is to introduce a multiplication on the set of homogeneous polynomials of fixed degree d, in a way to provide a duality theory between monomial ideals of K[x_1,\ldots,x_d] generated in degrees \leq n and block…

Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the…

For a subfield $\K$ of the field $\C$ of complex numbers, we consider curve and divisorial valuations on the algebra $\K[[x,y]]$ of formal power series in two variables with the coeficients in $\K$. We compute the semigroup Poincar\'e…

Let $\mathcal{C}_{\mathbi{d}},$ $\mathcal{I}_{\mathbi{d}},$ $\mathbi{d}{=}(d_1,d_2,..., d_n)$ be the algebras of join covariants and joint invariants of the $n$ binary forms of degrees $d_1,d_2,..., d_n.$ Formulas for computation of the…

Let $\R$ be a real closed field, $ {\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], $ with $ \deg_{Y}(Q) \leq 2, \deg_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m,$ and $ {\mathcal P} \subset \R[X_1,...,X_k] $ with $\deg_{X}(P)…

We propose a heuristic algorithm for fast computation of the Poincar\'{e} series $P_n(t)$ of the invariants of binary forms of degree $n$, viewed as rational functions. The algorithm is based on certain polynomial identities which remain to…

Earlier, there was computed the Poincar\'e series of a valuation or of a collection of valuations on the ring of germs of holomorphic functions in two variables. For a collection of several plane curve valuations it appeared to coincide…

A new explicit closed-form formula for the multivariate $(n, k)$th partial Bell polynomial $B_{n,k} (x_1, x_2, ..., x_{n - k + 1})$ is deduced. The formula involves multiple summations and makes it possible, for the first time, to easily…

Linear recursions of degree $k$ are determined by evaluating the sequence of Generalized Fibonacci Polynomials, $\{F_{k,n}(t_1,...,t_k)\}$ (isobaric reflects of the complete symmetric polynomials) at the integer vectors $(t_1,...,t_k)$. If…

We study permutation polynomials through the device of the \emph{polypermutation group} of an associative ring $R$, denoted by ${\rm Pgr}(R)$. We derive some basic properties and compute the cardinality of ${\rm Pgr}(\mathbb{Z}/p^k)$ when…

Let $\R$ be a real closed field, $ {\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], $ with $ \deg_{Y}(Q) \leq 2, \deg_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m$, and $ {\mathcal P} \subset \R[X_1,...,X_k] $ with $\deg_{X}(P)…

In this paper, we devise two algorithms for the problem of testing $q$-monomials of degree $k$ in any multivariate polynomial represented by a circuit, regardless of the primality of $q$. One is an $O^*(2^k)$ time randomized algorithm. The…