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

Let \{M_{r,s}\}_{0< r < p, 0< s < p'} be the irreducible Virasoro modules in the $(p,p')$-minimal series. In our previous paper, we have constructed a monomial basis of \oplus_{r=1}^{p-1}M_{r,s} in the case of $1<p'/p<2$. By `monomials' we…

We study the Poincar\'e series of the mixed and pure trace rings of generic matrices. These series are known to be rational functions. We obtain an explicit formula in lowest terms in the case of $2\times2$ matrices; a denominator, which we…

We study the set $\mathcal{L}_{F}$ of all $F$-vector spaces $L(P)$ where $P$ is monic and splits over $F$ and $L(Q)$ denotes the set of linear recurrence sequences over $F$ with characteristic polynomial $Q$. We show that $\mathcal{L}_{F}$…

We compute the algebraic K-theory of the non-commutative ring k<x_1,...,x_n>/(m^a) when k is a perfect field of positive characteristic and m=(x_1,...,x_n). We express the answer in terms of the truncation poset Witt vectors developed in…

We consider codes defined over an affine algebra $\mathcal A=R[X_1,\dots,X_r]/\left\langle t_1(X_1),\dots,t_r(X_r)\right\rangle$, where $t_i(X_i)$ is a monic univariate polynomial over a finite commutative chain ring $R$. Namely, we study…

A homogeneous set of monomials in a quotient of the polynomial ring $S:=F[x_1, \..., x_n]$ is called Gotzmann if the size of this set grows minimally when multiplied with the variables. We note that Gotzmann sets in the quotient $R:=F[x_1,…

For every finite Petri net, we construct a commutative polynomial in two variables and with coefficients from the semiring of natural numbers. We also present an inverse construction and show that multiplication of polynomials…

A class of generalized complex polynomials of Hermite type, suggested by a special magnetic Schrodinger operator, is introduced and some related basic properties are discussed.

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility…

In this paper, we present the compositional inverses of several classes permutation polynomials of the form $\sum_{i=1}^kb_i(x^{p^m}+x+\delta)^{s_i}-x$ over $\mathbb{F}_{p^{2m}}$, where for $1\leq i \leq k,$ $s_i, m$ are positive integers,…

Given a $d \times n$ integer matrix $A$, the main result is an elementary, simple-to-state algorithm that finds the largest $A$-graded ideal contained in any ideal $I$ in a polynomial ring $\Bbbk[x_1,\ldots,x_n]$. The special case where $A$…

Let S=K[X_1,...,X_n] be the polynomial ring over a field K. For bounded below Z^n-graded S-modules M and N we show that if Tor^S_p(M,N) is nonzero, then for every i between 0 and p, the dimension of the K-vector space Tor^S_i(M,N) is at…

In this paper, we consider Hermite and poly-Bernoulli mixed-type polynomials and investigate the properties of those polynomials which are derived from umbral calculus. Finally, we give various identities associated with Stirling numbers,…

We present an algorithm to decide whether a given ideal in the polynomial ring contains a monomial without using Gr\"obner bases, factorization or sub-resultant computations.

Let I be a monomial ideal of height c in a polynomial ring S over a field k. If I is not generated by a regular sequence, then we show that the sum of the betti numbers of S/I is at least 2^c + 2^{c-1} and characterize when equality holds.…

In this paper, we consider the degenerate multi-poly-Bernoulli numbers and polynomials which are defined by means of the multiple polylogarithms and degenerate versions of the multi-poly-Bernoulli numbers and polynomials. We investigate…

The notion of Frobenius Betti numbers generalizes the Hilbert-Kunz multiplicity theory and serves as an invariant that measures singularity. However, the explicit computation of the Frobenius Betti numbers of rings has been limited to very…

A number field $K$ is called \emph{monogenic} if its ring of integers $\mathbb{Z}_K$ can be expressed as a simple ring extension $\mathbb{Z}[\alpha]$ for some $\alpha \in \mathbb{Z}_K$. A monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$…

There exist several equivalent equations for the Poincar\'e series of a collection of valuations on the ring of germs of functions on a complex analytic variety. We give definitions of the Poinca\'e series of a collection of valuations in…

It is known that any $n$-variable function on a finite prime field of characteristic $p$ can be expressed as a polynomial over the same field with at most $p^n$ monomials. However, it is not obvious to determine the polynomial for a given…