交换代数

Over the past century, a correlation has been an essential mathematical technique utilized in engineering sciences, including practically every signal/image processing field. This paper describes an effective method of calculating the…

We describe a congruence property of solvable polynomials over Q, based on the irreducibility of cyclotomic polynomials over number fields that meet certain conditions.

We present a more general proof that cyclotomic polynomials are irreducible over Q and other number fields that meet certain conditions. The proof provides a new perspective that ties together well-known results, as well as some new…

We give a necessary and sufficient condition for an extension of valuation rings containing $\bf Q$ to be a filtered direct limit of smooth algebras.

We introduce a recursive decomposition algorithm for the Betti diagram of a complete intersection using the diagram of a complete intersection defined by a subset of the original generators. This alternative algorithm is the main tool that…

We propose a modification of the GPGCD algorithm, which has been presented in our previous research, for calculating approximate greatest common divisor (GCD) of more than 2 univariate polynomials with real coefficients and a given degree.…

We characterize all Gorenstein rings generated by strongly stable sets of monomials of degree two. We compute their Hilbert series in several cases, which also provides an answer to a question by Migliore and Nagel.

We introduce two new invariants of a Noetherian (standard graded) local ring $(R, \mathfrak m)$ that measure the number of generators of certain kinds of reductions of $\mathfrak m,$ and we study their properties. Explicitly, we consider…

Let $R$ be the Ehrhart ring of the stable set polytope of a cycle graph which is not Gorenstein. We describe the non-Gorenstein locus of $\mathrm{Spec} R$. Further, we show that $R$ is almost Gorenstein. Moreover, we show that the…

In this paper we explore consequences of the vanishing of ${\rm Ext}$ for finitely generated modules over a quasi-fiber product ring $R$; that is, $R$ is a local ring such that $R/(\underline x)$ is a non-trivial fiber product ring, for…

Inspired by the definition of balanced neighborly spheres, we define balanced neighborly polynomials and study the existence of these polynomials. The goal of this article is to construct balanced neighborly polynomials of type $(k,k,k,k)$…

We describe how one can explicitly obtain all atoms of an arbitrary root-closed monoid, whose quotient group is isomorphic to $\mathbb{Z}^2$. For this purpose, we solve this task for three special types of such monoids in Theorems 5 and 6,…

A toric ideal is called robust if its universal Gr\"obner basis is a minimal set of generators, and is called generalized robust if its universal Gr\"obner basis equals its universal Markov basis (the union of all its minimal sets of…

Motivated by results on the rationality of equivariant Hilbert series of some hierarchical models in algebraic statistics we introduce the Segre product of formal languages and apply it to establish rationality of equivariant Hilbert series…

The weakly Arf $(S_2)$-ification of a commutative Noetherian ring $R$ is considered to be a birational extension which is good next to the normalization. The weakly Arf property (WAP for short) of $R$ was introduced in 1971 by J. Lipman…

For any ideal $I$ of finite projective dimension in a commutative noetherian local ring $R$, we prove that if the conormal module $I/I^2$ has finite projective dimension over $R/I$, then $I$ must be generated by a regular sequence. This…

Let $K$ be a characteristic zero algebraic function field with a valuation $\nu$. Let $L$ be a finite extension of $K$ and $\omega$ be an extension of $\nu$ to $L$. We establish that the valuation ring $V_{\omega}$ of $\omega$ is…

This paper shows that Cohen-Macaulay algebras can be algebraically approximated in such a way that their Cohen-Macaulayness and minimal Betti numbers are preserved. This is achieved by showing that finitely generated modules over power…

The existence of certain Fq-spaces of differential forms of the projective line over a field K containing Fq leads us to prove an identity linking the determinant of the Moore matrix of n indeterminates with the determinant of the Moore…

We study conditions on polynomials such that the ideal generated by their orbits under the symmetric group action becomes a monomial ideal or has a monomial radical. If the polynomials are homogeneous, we expect that such an ideal has a…