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Some generalizations of strongly prime idealsAug 19 2019In this paper, we introduce the concepts of strongly 2-absorbing primary ideals (resp., submodules) and strongly 2-absorbing ideals (resp., submodules) as generalizations of strongly prime ideals. Furthermore, we investigate some basic properties of these ... More
Elasticity in Apery setsAug 18 2019A numerical semigroup $S$ is an additive subsemigroup of the non-negative integers, containing zero, with finite complement. Its multiplicity $m$ is its smallest nonzero element. The Apery set of $S$ is the set $\text{Ap}(S) = \{n \in S : n-m \notin S\}$. ... More
Cohomology of Burnside RingsAug 16 2019Let $G$ be a finite group and $A(G)$ its Burnside ring. For $H \subset G$ let $\mathbb{Z}_H$ denote the $A(G)$-module corresponding to the mark homomorphism associated to $H$. When the order of $G$ is square-free we give a complete description of the ... More
Laplacian algebras, manifold submetries and the Inverse Invariant Theory ProblemAug 15 2019Manifold submetries of the round sphere are a class of partitions of the round sphere that generalizes both singular Riemannian foliations, and the orbit decompositions by the orthogonal representations of compact groups. We exhibit a one-to-one correspondence ... More
A New Class of Irreducible PolynomialsAug 15 2019In this article, we propose a few sufficient conditions on polynomials having integer coefficients all of whose zeros lie outside the closed unit disc centered at the origin in the complex plane and deduce the irreducibility over the ring of integers. ... More
Edge rings of bipartite graphs with linear resolutionsAug 15 2019Ohsugi and Hibi characterized the edge ring of a finite connected simple graph with a $2$-linear resolution. On the other hand, Hibi, Matsuda and the author conjectured that the edge ring of a finite connected simple graph with a $q$-linear resolution, ... More
Invariants of polynomials mod Frobenius powersAug 14 2019Lewis, Reiner, and Stanton conjectured a Hilbert series for a space of invariants under an action of finite general linear groups using $(q,t)$-binomial coefficients. This work gives an analog in positive characteristic of theorems relating various Catalan ... More
Hilbert-Kunz Multiplicity of Fibers and Bertini TheoremsAug 13 2019Let $k$ be an algebraically closed field of characteristic $p > 0$. We show that if $X\subseteq\mathbb{P}^n_k$ is an equidimensional subscheme with Hilbert--Kunz multiplicity less than $\lambda$ at all points $x\in X$, then for a general hyperplane $H\subseteq\mathbb{P}^n_k$, ... More
The automorphism group of the zero-divisor digraph of matrices over an antiringAug 13 2019We determine the automorphism group of the zero-divisor digraph of the semiring of matrices over an antinegative commutative semiring with a finite number of zero-divisors.
RationalMaps, a package for Macaulay2Aug 12 2019This paper describes the RationalMaps package for Macaulay2. This package provides functionality for computing several aspects of rational maps such as whether a map is birational, or a closed embedding.
A note on partial coordinate system in a polynomial ringAug 12 2019J. Berson, J. W. Bikker and A. van den Essen proved that for a non-zerodivisor $a$ in a commutative ring $R$ containing $Q$ if the polynomials $f_1,\dots,f_{n-1}$ in $R[X_1,\dots,X_n]$ form a partial coordinate system over the rings $R_a$ and $\dfrac{R}{aR}$ ... More
Schemes supported on the singular locus of a hyperplane arrangement in $\mathbb P^n$Aug 11 2019We introduce the use of liaison addition to the study of hyperplane arrangements. For an arrangement, $\mathcal A$, of hyperplanes in $\mathbb P^n$, $\mathcal A$ is free if $R/J$ is Cohen-Macaulay, where $J$ is the Jacobian ideal of $\mathcal A$. Terao's ... More
Log canonical thresholds of generic links of determinantal varietiesAug 11 2019We show that the log canonical threshold of a generic determinantal variety and its generic link are the same.
The edge ideal of a graph and its split graphsAug 10 2019We introduce and study the concept which we call the splitting of a graph and compare algebraic properties of the edge ideals of graphs and those of their splitting graphs.
On the monomial reduction number of a monomial ideal in $K[x,y]$Aug 10 2019The reduction number of monomial ideals in the polynomial $K[x,y]$ is studied. We focus on ideals $I$ for which $J=(x^a,y^b)$ is a reduction ideal. The computation of the reduction number amounts to solve linear inequalities. In some special cases the ... More
Symmetry, Unimodality, and Lefschetz Properties for Graded ModulesAug 09 2019If $\mbk$ is algebraically closed of characteristic zero and $R = \mbk[x,y, z]$, we first investigate the Weak Lefschetz Property for the finite length $R$-module $M$ that is the cokernel of a map $\vp: \bds_{j=1}^{n+2} R(-b_j)\to\bds_{i=1}^n R(-a_i)$. ... More
On Residual and Stable CoordinatesAug 09 2019In a recent paper, M. E. Kahoui and M. Ouali have proved that over an algebraically closed field $k$ of characteristic zero, residual coordinates in $k[X][Z_1,\dots,Z_n]$ are one-stable coordinates. In this paper we extend their result to the case of ... More
Pretorsion theories in general categoriesAug 09 2019We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair $(\mathcal T, \mathcal F)$ of full replete subcategories in a category $\mathcal C$, the corresponding full subcategory $\mathcal Z = ... More
On Double Danielewski Surfaces and the Cancellation ProblemAug 09 2019We study a two-dimensional family of affine surfaces which are counter-examples to the Cancellation Problem. We describe the Makar-Limanov invariant of these surfaces, determine their isomorphism classes and characterize the automorphisms of these surfaces. ... More
A localized version of the basic triangle theoremAug 09 2019In this short note, we give a localized version of the basic triangle theorem, first published in 2011 (see [4]) in order to prove the independence of hyperlogarithms over various function fields. This version provides direct access to rings of scalars ... More
Regularity of Edge Ideals Via SuspensionAug 08 2019We study the Castelnuovo-Mumford regularity of powers of edge ideals. We prove that if G is a bipartite graph, then reg(I(G)^s) \leq 2s + reg I(G) - 2 for all s \geq 2, which is the best possible upper bound for any s. Suspension plays a key role in proof ... More
Subgroups of an abelian group, related ideals of the group ring, and quotients by those idealsAug 08 2019Let $RG$ be the group ring of an abelian group $G$ over a commutative ring $R$ with identity. An injection $\Phi$ from the subgroups of $G$ to the non-unit ideals of $RG$ is well-known. It is defined by $\Phi(N)=I(R,N)RG$ where $I(R,N)$ is the augmentation ... More
A conjecture on cluster automorphisms of cluster algebrasAug 08 2019A cluster automorphism is a $\mathbb{Z}$-algebra automorphism of a cluster algebra $\mathcal A$ satisfying that it sends a cluster to another and commutes with mutations. Chang and Schiffler conjectured that a cluster automorphism of $\mathcal A$ is just ... More
Green-Lazarsfeld Condition for Toric Edge Ideals of Bipartite GraphsAug 07 2019Previously, Ohsugi and Hibi gave a combinatorial description of bipartite graphs $G$ whose toric edge ideal $I_G$ is generated by quadrics, showing that every cycle of $G$ of length at least $6$ must have a chord. This corresponds to the Green-Lazarsfeld ... More
On an example concerning the second rigidity theoremAug 06 2019In this paper we revisit an example of Celikbas and Takahashi concerning the reflexivity of tensor products of modules. We study Tor-rigidity and the Hochster--Huneke graph with vertices consisting of minimal prime ideals, and determine a condition with ... More
Multigraded Shifts of Matroidal IdealsAug 06 2019In this paper, we show that if $I$ is a matroidal ideal, then the ideal generated by the $i$-th multigraded shifts is also a matroidal ideal for every $i=0,\ldots,\text{pd}(I)$.
Implicitization of tensor product surfaces via virtual projective resolutionsAug 06 2019We derive the implicit equations for certain parametric surfaces in three-dimensional projective space termed tensor product surfaces. Our method computes the implicit equation for such a surface based on the knowledge of the syzygies of the base point ... More
Symbolic powers of monomial idealsAug 06 2019Let $A = K[X_1,\ldots, X_d]$ and let $I$, $J$ be monomial ideals in $A$. Let $I_n(J) = (I^n \colon J^\infty)$ be the $n^{th}$ symbolic power of $I$ \wrt \ $J$. It is easy to see that the function $f^I_J(n) = e_0(I_n(J)/I^n)$ is of quasi-polynomial type, ... More
On the $k$-torsion of the module of differentials of order $n$ of hypersurfacesAug 05 2019We characterize the $k$-torsion freeness of the module of differentials of order $n$ of a point of a hypersurface in terms of the singular locus of the corresponding local ring.
Covers of rational double points in mixed characteristicAug 04 2019We further the classification of rational surface singularities. Suppose $(S, \mathfrak{n}, \mathcal{k})$ is a strictly Henselian regular local ring of mixed characteristic $(0, p > 5)$. We classify functions $f$ for which $S/(f)$ has an isolated rational ... More
Toric ideals of Minkowski sums of unit simplicesAug 04 2019In this paper, we discuss the toric ideals of Minkowski sums of unit simplices. More precisely, we prove that the toric ideal of Minkowski sum of unit simplices has a squarefree initial ideal and is generated by quadratic binomials. Moreover, we also ... More
Nef-partitions arising from unimodular configurationsAug 04 2019Reflexive polytopes have been studied from viewpoints of combinatorics, commutative algebra and algebraic geometry. A nef-partition of a reflexive polytope $\mathcal{P}$ is a decomposition $\mathcal{P}=\mathcal{P}_1+\cdots+\mathcal{P}_r$ such that each ... More
Bounds of the multiplicity of abelian quotient complete intersection singularitiesAug 03 2019In this paper, we investigate the multiplicities and the log canonical thresholds of abelian quotient complete intersection singularities in term of the special datum. Moreover we give bounds of the multiplicity of abelian quotient complete intersection ... More
Homogeneous Liaison and the Sequentially Bounded Licci PropertyAug 02 2019In CI-Liaison, significant effort has been made to study ideals that are in the linkage class of a complete intersection, which are called licci ideals. In a polynomial ring, recently E. Chong defined a "sequentially bounded" condition on the degrees ... More
Hilbert schemes with few Borel fixed pointsJul 31 2019Aug 02 2019We characterize Hilbert polynomials that give rise to Hilbert schemes with two Borel fixed points and determine when the associated Hilbert schemes or its irreducible components are non-singular. More generally, we show that the Hilbert scheme is reduced ... More
Hilbert schemes with few Borel fixed pointsJul 31 2019We characterize Hilbert polynomials that give rise to Hilbert schemes with two Borel fixed points and determine when the associated Hilbert schemes or its irreducible components are non-singular. More generally, we show that the Hilbert scheme is reduced ... More
On the rigidity of certain Pham-Brieskorn ringsJul 30 2019Fix a field $k$ of characteristic zero. If $a_1, ..., a_n$ ($n>2$) are positive integers, the integral domain $B = k[X_1, ..., X_n] / ( X_1^{a_1} + ... + X_n^{a_n} )$ is called a Pham-Brieskorn ring. It is conjectured that if $a_i > 1$ for all $i$ and ... More
On Semisimple SemiringsJul 30 2019We investigate ideal-semisimple and congruence-semisimple semirings. We give several new characterizations of such semirings using e-projective and e-injective semimodules. We extend several characterizations of semisimple rings to (not necessarily subtractive) ... More
Projective toric codes over hypersimplicesJul 30 2019Aug 16 2019Let $d\geq 1$ be an integer, and let $\mathcal{P}$ be the convex hull in $\mathbb{R}^s$ of all integral points $\mathbf{e}_{i_1}+\cdots+\mathbf{e}_{i_d}$ such that $1\leq i_1<\cdots< i_d\leq s$, where $\mathbf{e}_i$ is the $i$-th unit vector in $\mathbb{R}^s$. ... More
A note on projective toric codesJul 30 2019Let $d\geq 1$ be an integer, and let $\mathcal{P}$ be the convex hull in $\mathbb{R}^s$ of all integral points $\mathbf{e}_{i_1}+\cdots+\mathbf{e}_{i_d}$ such that $1\leq i_1<\cdots< i_d\leq s$, where $\mathbf{e}_i$ is the $i$-th unit vector in $\mathbb{R}^s$. ... More
High dimensional affine codes whose square has a designed minimum distanceJul 30 2019Given a linear code $\mathcal{C}$, its square code $\mathcal{C}^{(2)}$ is the span of all component-wise products of two elements of $\mathcal{C}$. Motivated by applications in multi-party computation, our purpose with this work is to answer the following ... More
Invariant rings and representations of the symmetric groupsJul 30 2019In this paper we study invariant rings arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form $K[U]^{\Gamma}$ where $\Gamma$ is a product of general linear groups over a field $K$ of characteristic ... More
The local cohomology of a parameter ideal with respect to an arbitrary idealJul 30 2019Let $S$ be a complete intersection presented as $R/J$ for $R$ a regular ring and $J$ a parameter ideal in $R$. Let $I\subseteq R$ be an ideal containing $J$, corresponding to an arbitrary ideal of $S$. It is well known that the set of associated primes ... More
Depth of an initial idealJul 30 2019Given an arbitrary integer $d>0$, we construct a homogeneous ideal $I$ of the polynomial ring $S = K[x_1, \ldots, x_{3d}]$ in $3d$ variables over a filed $K$ for which $S/I$ is a Cohen--Macaulay ring of dimension $d$ with the property that, for each of ... More
Depth of an initial idealJul 30 2019Aug 01 2019Given an arbitrary integer $d>0$, we construct a homogeneous ideal $I$ of the polynomial ring $S = K[x_1, \ldots, x_{3d}]$ in $3d$ variables over a filed $K$ for which $S/I$ is a Cohen--Macaulay ring of dimension $d$ with the property that, for each of ... More
On certain polynomial systems involving Stirling numbers of second kindJul 29 2019We solve a special type of linear systems with coefficients in multivariate polynomial rings. These systems arise in the computation of parametric Bernstein-Sato polynomials associated with certain hypergeometric ideals in the Weyl algebra.
Algebraic $h$-vectors of simplicial complexes through local cohomology, part 1Jul 29 2019Given an infinite field $\mathbb{k}$ and a simplicial complex $\Delta$, a common theme in studying the $f$- and $h$-vectors of $\Delta$ has been the consideration of the Hilbert series of the Stanley--Reisner ring $\mathbb{k}[\Delta]$ modulo a generic ... More
On the Gorensetein $(n,d)$-Flat and Gorensetein $(n,d)$-Injective ModulesJul 29 2019Let R be a ring. In this paper, Gorenstein (n,d)-flat and (n,d)-Gorenstein (n,d)-injective modules and some of their basic properties are studied. Moreover, some characterizations of rings over Gorenstein (n,d)-flat and (n,d)-Gorenstein (n,d)-injective ... More
On local real algebraic geometry and applications to kinematicsJul 28 2019We address the question of identifying non-smooth points in affine real algebraic varieties. A simple algebraic criterion will be formulated and proven. As an application we can answer several questions about the configuration spaces of some class of ... More
On local real algebraic geometry and applications to kinematicsJul 28 2019Aug 06 2019We address the question of identifying non-smooth points in affine real algebraic varieties. A simple algebraic criterion will be formulated and proven. As an application we can answer several questions about the configuration spaces of some class of ... More
Squarefree monomial ideals with maximal depthJul 27 2019Let $(R,\mm)$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We say $M$ has maximal depth if there is an associated prime $\pp$ of $M$ such that $\depth M=\dim R/\pp$. In this paper we study squarefree monomial ideals which have maximal ... More
Cohen-Macaulay local rings with $e_1 = e + 2$Jul 26 2019In this paper we determine the possible Hilbert functions of a Cohen-Macaulay local ring of dimension $d$, multiplicity $e$ and first Hilbert coefficient $e_1$ in the case $e_1 = e + 2$.
Castelnuovo-Mumford regularity and related invariantsJul 26 2019These notes are an introduction to some basic aspects of the Castelnuovo-Mumford regularity and related topics such as weak regularity, a*-invariant and partial regularities.
Bernstein-Sato roots for monomial ideals in prime characteristicJul 26 2019Following work of Musta\c{t}\u{a} and Bitoun we recently developed a notion of Bernstein-Sato roots for arbitrary ideals, which is a prime characteristic analogue for the roots of the Bernstein-Sato polynomial. Here we prove that for monomial ideals the ... More
Relative Generalized Minimum Distance FunctionJul 25 2019In this paper we introduce the relative generalized minimum distance function (RGMDF for short) and it allows us to give an algebraic approach to the relative generalized Hamming weights of the projective Reed--Muller--type codes. Also we introduce the ... More
Telescope conjecture for homotopically smashing t-structures over commutative noetherian ringsJul 25 2019We show that any homotopically smashing t-structure in the derived category of a commutative noetherian ring is compactly generated. This generalizes the validity of the telescope conjecture for commutative noetherian rings due to Neeman. As another consequence, ... More
An upper bound on the first homology of spline complexesJul 25 2019Let $\Delta$ be a connected, pure $2$-dimensional simplicial complex embedded in $\mathbb{R}^2$ and let $C^{r}(\hat{\Delta})$ be the homogenized spline module of $\Delta$ with smoothness $r$. To study $C^{r}(\hat{\Delta})$, Schenck and Stillman developed ... More
Associated primes and integral closure of Noetherian ringsJul 24 2019Let A be a Noetherian ring and B be a finitely generated A-algebra. Denote by A' the integral closure of A in B. We give necessary and sufficient conditions for a prime p in A to be in Ass_{A}(B/A') generalizing and strengthening classical results for ... More
Semigroups, Projections, $κ$-domainsJul 24 2019We propose an extension of the classical notion of projection to semigroups and provide conditions under which a semigroup embeds in a complete lattice. We also introduce the new notion of a $\kappa$-domain and prove some useful separation results valid ... More
The norm of the saturation of a binomial ideal, and applications to Markov basesJul 24 2019Given a pure binomial ideal I in variables x_i, we define a new measure of the complexity of the saturation of I with respect to the product of the variables x_i, which we call the norm. We give a bound on the norm in terms of easily-computed invariants ... More
Bernstein-Sato functional equations, $V$-filtrations, and multiplier ideals of direct summandsJul 23 2019Jul 24 2019This paper investigates the existence and properties of a Bernstein-Sato functional equation in nonregular settings. In particular, we construct $D$-modules in which such formal equations can be studied. The existence of the Bernstein-Sato polynomial ... More
Bernstein-Sato functional equations, $V$-filtrations, and multiplier ideals of direct summandsJul 23 2019This paper investigates the existence and properties of a Bernstein-Sato functional equation in nonregular settings. In particular, we construct $D$-modules in which such formal equations can be studied. The existence of the Bernstein-Sato polynomial ... More
Annihilators of $D$-modules in mixed characteristicJul 23 2019Let $R$ be a polynomial or formal power series ring with coefficients in a DVR $V$ of mixed characteristic with a uniformizer $\pi$. We prove that the $R$-module annihilator of any nonzero $\D(R,V)$-module is either zero or is generated by a power of ... More
Factorization Theory in Commutative MonoidsJul 23 2019This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative ... More
Approximation results of Artin-Tougeron-type for general filtrations and for $C^r$-equationsJul 23 2019Artin approximation and other related approximation results are used in various areas. The traditional formulation of such results is restricted to filtrations by powers of ideals, $\{I^j\}$, and to Noetherian rings. In this short note we extend several ... More
Surjectivity of the completion map for rings of $C^\infty$-functions. Necessary conditions and sufficient conditionsJul 23 2019Consider the ring of smooth function germs at the origin of $\mathbb{R}^n$. The Taylor expansion is the completion map from this ring to the ring of formal power series. Borel's lemma ensures the surjectivity of this map. In this short note we address ... More
Betti tables of monomial ideals fixed by permutations of the variablesJul 23 2019Let $S_n$ be a polynomial ring with $n$ variables over a field and $\{I_n\}_{n \geq 1}$ a chain of ideals such that each $I_n$ is a monomial ideal of $S_n$ fixed by permutations of the variables. In this paper, we present a way to determine all nonzero ... More
A formula for systems of Boolean polynomial equations and applications to parametrized complexityJul 23 2019It is known a method for transforming a system of Boolean polynomial equations to a single Boolean polynomial equation with less variables. In this paper, we improve the method, and give a formula in the Boolean polynomial ring for systems of Boolean ... More
A formula for systems of Boolean polynomial equations and applications to computational complexityJul 23 2019Jul 26 2019It is known a method for transforming a system of Boolean polynomial equations to a single Boolean polynomial equation with less variables. In this paper, we improve the method, and give a formula in the Boolean polynomial ring for systems of Boolean ... More
On the colength of fractional idealsJul 23 2019The main result in this paper is to supply a recursive formula, on the number of minimal primes, for the colength of a fractional ideal in terms of the maximal points of the value set of the ideal itself. The fractional ideals are taken in the class of ... More
Hamming Polynomial of a DemimatroidJul 23 2019Following Britz, Johnsen, Mayhew and Shiromoto, we consider demi\-ma\-troids as a(nother) natural generalization of matroids. As they have shown, demi\-ma\-troids are the appropriate combinatorial objects for studying Wei's duality. Our results here apport ... More
Versal deformations of pairs and Cohen-Macaulay approximationJul 21 2019For a pair (algebra, module) with isolated singularity we establish the existence of a versal henselian deformation. Obstruction theory in terms of an Andr\'e-Quillen cohomology for pairs is a central ingredient in the Artin theory used. Cohen-Macaulay ... More
Versal deformations of pairs and Cohen-Macaulay approximationJul 21 2019Jul 25 2019For a pair (algebra, module) with isolated singularity we establish the existence of a versal henselian deformation. Obstruction theory in terms of an Andr\'e-Quillen cohomology for pairs is a central ingredient in the Artin theory used. Cohen-Macaulay ... More
Deformation theory of Cohen-Macaulay approximationJul 21 2019Jul 25 2019In a previous article (J. Algebra 367 (2012), 142-165) we established axiomatic parametrised Cohen-Macaulay approximation which in particular was applied to pairs consisting of a finite type flat family of Cohen-Macaulay rings and modules. In this sequel ... More
Deformation theory of Cohen-Macaulay approximationJul 21 2019In a previous article (J. Algebra 367 (2012), 142-165) we established axiomatic parametrised Cohen-Macaulay approximation which in particular was applied to pairs consisting of a finite type flat family of Cohen-Macaulay rings and modules. In this sequel ... More
Integrality of matrices, finiteness of matrix semigroups, and dynamics of linear cellular automataJul 19 2019Let $\mathbb{K}$ be a finite commutative ring, and let $\mathbb{L}$ be a commutative $\mathbb{K}$-algebra. Let $A$ and $B$ be two $n \times n$-matrices over $\mathbb{L}$ that have the same characteristic polynomial. The main result of this paper states ... More
Primality of multiply connected polyominoesJul 19 2019Jul 26 2019It is known that the polyomino ideal of simple polyominoes is prime. In this paper, we focus on multiply connected polyominoes, namely polyominoes with holes, and observe that the non-existence of a certain sequence of inner intervals of the polyomino, ... More
Primality of multiply connected polyominoesJul 19 2019It is known that the polyomino ideal of simple polyominoes is prime. In this paper, we focus on multiply connected polyominoes, namely polyominoes with holes, and observe that the non-existence of a certain sequence of inner intervals of the polyomino, ... More
Asymptotic Lech's inequalityJul 19 2019We explore the classical Lech's inequality relating the Hilbert--Samuel multiplicity and colength of an $\mathfrak{m}$-primary ideal in a Noetherian local ring $(R,\mathfrak{m})$. We prove optimal versions of Lech's inequality for sufficiently deep ideals ... More
The structure and free resolutions of the symbolic powers of star configurations of hypersurfacesJul 18 2019Aug 05 2019Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from ... More
The structure and free resolution of the symbolic powers of star configurations of hypersurfacesJul 18 2019Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from ... More
Irreducible Generalized Numerical Semigroups and uniqueness of the Frobenius elementJul 18 2019Let $\mathbb{N}^{d}$ be the $d$-dimensional monoid of non-negative integers. A generalized numerical semigroup is a submonoid $ S\subseteq \mathbb{N}^d$ such that $H(S)=\mathbb{N}^d \setminus S$ is a finite set. We introduce irreducible generalized numerical ... More
The Geometric Syzygy Conjecture in Even GenusJul 17 2019Aug 05 2019We prove the Geometric Syzygy Conjecture for generic canonical curves of even genus. This result extends Green's classical result on the generation of the ideal of a canonical curve by rank four quadrics to the highest linear syzygy group.
The Geometric Syzygy Conjecture in Even GenusJul 17 2019We prove the Geometric Syzygy Conjecture for generic canonical curves of even genus.
On the radius of the category of extensions of matrix factorizationsJul 17 2019Let $S$ be a commutative noetherian ring. The extensions of matrix factorizations of non-zerodivisors $x_1,\dots,x_n$ of $S$ form a full subcategory of finitely generated modules over the quotient ring $S/(x_1\cdots x_n)$. In this paper, we investigate ... More
Bernstein-Sato theory for arbitrary ideals in positive characteristicJul 17 2019Musta\c{t}\u{a} defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the $F$-jumping numbers of the ideal. This approach was later refined by Bitoun. Here we generalize ... More
Graded Cohen-Macaulay domains and lattice polytopes with short $h$-vectorJul 16 2019Let P be a lattice polytope with $h^*$-vector $(1, h^*_1, h^*_2)$. In this note we show that 1) if $h_2^* \leq h_1^*$, then $P$ is IDP, and 2) if $h_2^* \leq h_1^* - 1$, then $P$ is Koszul (in characteristic 0). More generally, we show the corresponding ... More
Graded Cohen-Macaulay domains and lattice polytopes with short $h$-vectorJul 16 2019Jul 18 2019Let P be a lattice polytope with $h^*$-vector $(1, h^*_1, h^*_2)$. In this note we show that 1) if $h_2^* \leq h_1^*$, then $P$ is IDP, and 2) if $h_2^* \leq h_1^* - 1$, then $P$ is Koszul (in characteristic 0). More generally, we show the corresponding ... More
Dedekind semidomainsJul 16 2019We define Dedekind semidomains as semirings in which each nonzero fractional ideal is invertible. Then we find some equivalent condition for semirings to being Dedekind. For example, we prove that a Noetherian semidomain is Dedekind if and only if it ... More
Symbolic analytic spread: upper bounds and applicationsJul 16 2019The symbolic analytic spread of an ideal $I$ is defined in terms of the rate of growth of the minimal number of generators of its symbolic powers. In this article we find upper bounds for the symbolic analytic spread under certain conditions in terms ... More
What is... a Markov basis?Jul 16 2019This short piece defines a Markov basis. The aim is to introduce the statistical concept to mathematicians.
Almost normally torsionfree idealsJul 15 2019We describe all connected graphs whose edge ideals are almost normally torsionfree. We also prove that the facet ideal of a special odd cycle is almost normally torsionfree. Finally, we determine the t-spread principal Borel ideals generated in degree ... More
Depth functions of symbolic powers of homogeneous idealsJul 15 2019This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function depth R/I^(t) = dim R - pd I^(t) - 1, where I^(t) denotes the t-th symbolic power of a ... More
Depth and detection for Noetherian unstable algebrasJul 15 2019For a connected Noetherian unstable algebra $R$ over the mod $p$ Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of $R$, originally proved when $R$ is the mod $p$ cohomology ring of a finite group. This recovers the ... More
Flat Semimodules & von Neumann Regular SemiringsJul 13 2019Flat modules play an important role in the study of the category of modules over rings and in the characterization of some classes of rings. We study the e-flatness for semimodules introduced by the first author using his new notion of exact sequences ... More
Integrality over ideal semifiltrationsJul 13 2019We study integrality over rings (all commutative in this paper) and over ideal semifiltrations (a generalization of integrality over ideals). We begin by reproving classical results, such as a version of the "faithful module" criterion for integrality ... More
An introduction to Mathieu subspacesJul 13 2019This is the note for the four lectures given by the author in the ``International Short-School/Conference on Affine Algebraic Geometry and the Jacobian Conjecture" at Chern Institute of Mathematics, Nankai University, Tianjin, China. July 14-25, 2014. ... More
Mathieu-Zhao spaces of polynomial ringsJul 13 2019We describe all Mathieu-Zhao spaces of $k[x_1,\cdots,x_n]$ ($k$ is an algebraically closed field of characteristic zero) which contains an ideal of finite codimension. Furthermore we give an algorithm to decide if a subspace of the form $I+kv_1+\cdots+kv_r$ ... More
Cotilting with balanced big Cohen-Macaualay modulesJul 12 2019Over a Cohen-Macaulay local ring admitting a canonical module the definable closure of the class of balanced big Cohen-Macaulay modules is cotilting and is the smallest such class containing the maximal Cohen-Macaulay modules. We describe its cotilting ... More
Good rings and homogeneous polynomialsJul 12 2019In 2011, Khurana, Lam and Wang define the following property. (*)A commutative unital ring A satisfies the property "power stable range one" if for all a, b $\in$ A with aA + bA = A there is an integer N = N (a, b) $\ge$ 1, $\lambda$ = $\lambda$(a, b) ... More