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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 in new cases. To this end we show that the Segre product of two regular languages is again regular. We also prove that every filtration of algebras given as a tensor product of families of algebras with rational equivariant Hilbert series has a rational equivariant Hilbert series. The term equivariant is used broadly to include the action of the monoid of nonnegative integers by shifting variables. Furthermore, we exhibit a filtration of shift invariant monomial algebras that has a rational equivariant Hilbert series, but whose presentation ideals do not stabilize., 26 pages, comments are welcome!

\emph{Proto-exact categories}, introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a \emph{non-additive} setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor. In this paper, we show that the categories of modules over semirings and hyperrings - algebraic structures which have gained prominence in tropical geometry - carry proto-exact structures. In the first part, we prove that the category of modules over a semiring is equipped with a proto-exact structure; modules over an idempotent semiring have a strong connection to matroids. We also prove that the category of algebraic lattices $\mathcal{L}$ has a proto-exact structure, and furthermore that the subcategory of $\mathcal{L}$ consisting of finite lattices is equivalent to the category of finite $\mathbb{B}$-modules as proto-exact categories, where $\mathbb{B}$ is the \emph{Boolean semifield}. We also discuss some relations between $\mathcal{L}$ and geometric lattices (simple matroids) from this perspective. In the second part, we prove that the category of modules over a hyperring has a proto-exact structure. In the case of finite modules over the \emph{Krasner hyperfield} $\mathbb{K}$, a well-known relation between finite $\mathbb{K}$-modules and finite incidence geometries yields a combinatorial interpretation of exact sequences.

It has been recently observed that fundamental aspects of the classical theory of factorization can be greatly generalized by combining the languages of monoids and preorders. This has led to various theorems on the existence of certain factorizations, herein called $\preceq$-factorizations, for the $\preceq$-non-units of a (multiplicatively written) monoid $H$ endowed with a preorder $\preceq$, where an element $u \in H$ is a $\preceq$-unit if $u \preceq 1_H \preceq u$ and a $\preceq$-non-unit otherwise. The ``building blocks'' of these factorizations are the $\preceq$-irreducibles of $H$ (i.e., the $\preceq$-non-units $a \in H$ that cannot be written as a product of two $\preceq$-non-units each of which is strictly $\preceq$-smaller than $a$); and it is interesting to look for sufficient conditions for the $\preceq$-factorizations of a $\preceq$-non-unit to be bounded in length or finite in number (if measured or counted in a suitable way). This is precisely the kind of questions addressed in the present work, whose main novelty is the study of the interaction between minimal $\preceq$-factorizations (i.e., a refinement of $\preceq$-factorizations used to counter the ``blow-up phenomena'' that are inherent to factorization in non-commutative or non-cancellative monoids) and some finiteness conditions describing the ``local behaviour'' of the pair $(H, \preceq)$. Besides a number of examples and remarks, the paper includes many arithmetic results, a part of which are new already in the basic case where $\preceq$ is the divisibility preorder on $H$ (and hence in the setup of the classical theory)., Comment: 27 pages, 2 figures. Final version to appear in Journal of Algebra

We introduce the notion of residual intersections of modules and prove their existence. We show that projective dimension one modules have Cohen-Macaulay residual intersections, namely they satisfy the relevant Artin-Nagata property. We then establish a formula for the core of orientable modules satisfying certain homological conditions, extending previous results of Corso, Polini, and Ulrich on the core of projective one modules. Finally, we provide examples of classes of modules that satisfy our assumptions.

A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen-Macaulayness of absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem is whether the splinter property spreads from a point to an open neighborhood of a noetherian scheme. Our paper addresses this problem in prime characteristic, where we show that a locally noetherian scheme that has finite Frobenius or that is locally essentially of finite type over a quasi-excellent local ring has an open splinter locus. In particular, all varieties over fields of positive characteristic have open splinter loci. Intimate connections are established between the openness of splinter loci and $F$-compatible ideals, which are prime characteristic analogues of log canonical centers. We prove the surprising fact that for a large class of noetherian rings with pure (aka universally injective) Frobenius, the splinter condition is detected by the splitting of a single generically \'etale finite extension. We also show that for a noetherian $\textbf{N}$-graded ring over a field, the homogeneous maximal ideal detects the splinter property., Comment: Comments welcome, minor updates in v2

Let $M$ be a module over a Noetherian local ring $A$. We study $M$-independent sequences of elements of $\mathfrak{m}_A$ in the sense of Lech and Hanes. The main tool is a new characterization of the $M$-independence of a sequence in terms of the associated Koszul complex. As applications, we give a result in linkage theory, a freeness criterion for $M$ in terms of the existence of a strongly $M$-independent sequence of length $edim(A)$, and another freeness criterion inspired from the patching method of Calegari and Geraghty for balanced modules in their 2018 paper., 17 pages

This paper describes the centers of the universal enveloping algebras and the invariant rings of the standard filiform Lie algebras over fields of characteristic zero and also over large enough prime characteristic. We determine explicit generators for the quotient fields and also a compact form for the generators for the invariants rings. We prove several combinatorial results concerning the Hilbert series of these algebras.

In this paper, we consider the homogeneous coordinate rings $A(Y_{n,d}) \cong \mathbb{K}[\Omega_{n,d}]$ of monomial projections $Y_{n,d}$ of Veronese varieties parameterized by subsets $\Omega_{n,d}$ of monomials of degree $d$ in $n+1$ variables where: (1) $\Omega_{n,d}$ contains all monomials supported in at most $s$ variables and, (2) $\Omega_{n,d}$ is a set of monomial invariants of a finite diagonal abelian group $G \subset GL(n+1,\mathbb{K})$ of order $d$. Our goal is to study when $\mathbb{K}[\Omega_{n,d}]$ is a quadratic algebra and, if so, when $\mathbb{K}[\Omega_{n,d}]$ is Koszul or G-quadratic. For the family (1), we prove that $\mathbb{K}[\Omega_{n,d}]$ is quadratic when $s \ge \lceil \frac{n+2}{2} \rceil$. For the family (2), we completely characterize when $\mathbb{K}[\Omega_{2,d}]$ is quadratic in terms of the group $G \subset GL(3,\mathbb{K})$, and we prove that $\mathbb{K}[\Omega_{2,d}]$ is quadratic if and only if it is Koszul. We also provide large families of examples where $\mathbb{K}[\Omega_{n,d}]$ is G-quadratic., Comment: To appear in Journal of Algebra

Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of $A$, we show that $A$ is almost strong Lefschetz. This quite modest result is nevertheless arguably the most encompassing so far concerning the strong Lefschetz property for graded codimension three AG algebras.

In this paper, we give the necessary and sufficient conditions for the Cohen-Macaulayness of the associated graded ring of a simplicial affine semigroups using Gr\"{o}bner basis. We generalize the concept of homogeneous numerical semigroup for the simplicial affine semigroup and show that the Betti numbers of the corresponding semigroup ring matches with the Betti numbers of the associated graded ring. We also define the nice extension for simplicial affine semigroups, motivated by the notion of a nice extension of the numerical semigroups.

Let $\mathbb{K}$ be a field and $I$ be a square-free monomial ideal in the polynomial ring $\mathbb{K}[x_1, \ldots, x_n]$. The Green-Lazarsfeld index, $\mathrm{index}(I)$, counts the number of steps to reach to a syzygy minimally generated by a nonlinear form in a graded minimal free resolution of $I$. In this paper, we study this invariant for $I$ and its powers from a combinatorial point of view. We characterize all square-free monomial ideals $I$ generated in degree $3$ such that $\mathrm{index}(I)>1$. Utilizing this result, we also characterize all square-free monomial ideals generated in degree $3$ such that $\mathrm{index}(I)>1$ and $\mathrm{index}(I^2)=1$. In case $n\leq5$, it is shown that $\mathrm{index}(I^k)>1$ for all $k$ if $I$ is any square-free monomial ideal with $\mathrm{index}(I)>1$., 14 pages, 4 figures

We show that a sufficiently general hypersurface of degree $d$ in $n$-dimensional projective space admits a toric Gr\"obner degeneration if and only if $d\leq 2n-1$., Comment: 6 pages; v2 minor revisions. To appear in Canadian Mathematical Bulletin

Given a simple undirected graph $G$ there is a simplicial complex $\mathrm{Ind}(G)$, called the independence complex, whose faces correspond to the independent sets of $G$. This is a well studied concept because it provides a fertile ground for interactions between commutative algebra, graph theory and algebraic topology. One of the line of research pursued by many authors is to determine the graph classes for which the associated independence complex is Cohen-Macaulay. For example, it is known that when $G$ is a chordal graph the complex $\mathrm{Ind}(G)$ is in fact vertex decomposable, the strongest condition in the Cohen-Macaulay ladder. In this article we consider a generalization of independence complex. Given $r\geq 1$, a subset of the vertex set is called $r$-independent if the connected components of the induced subgraph have cardinality at most $r$. The collection of all $r$-independent subsets of $G$ form a simplicial complex called the $r$-independence complex and is denoted by $\mathrm{Ind}_r(G)$. It is known that when $G$ is a chordal graph the complex $\mathrm{Ind}_r(G)$ has the homotopy type of a wedge of spheres. Hence it is natural to ask which of these complexes are shellable or even vertex decomposable. We prove, using Woodroofe's chordal hypergraph notion, that these complexes are always shellable when the underlying chordal graph is a tree. Further, using the notion of vertex splittable ideals we show that for caterpillar graphs the associated $r$-independence complex is vertex decomposable for all values of $r$. We also construct chordal graphs on $2r+2$ vertices such that their $r$-independence complexes are not sequentially Cohen-Macaulay for any $r \ge 2$., Final version. To appear in International Journal of Algebra and Computation

Let $S=\mathbb{K}[x_1,\ldots, x_n]$ be the polynomial ring over a field $\mathbb{K}$ and $\mathfrak{m}= (x_1, \ldots, x_n)$ be the irredundant maximal ideal of $S$. For an ideal $I \subset S$, let $\mathrm{sat}(I)$ be the minimum number $k$ for which $I \colon \mathfrak{m}^k = I \colon \mathfrak{m}^{k+1}$. In this paper, we compute the saturation number of irreducible monomial ideals and their powers. We apply this result to find the saturation number of the ordinary powers and symbolic powers of some families of monomial ideals in terms of the saturation number of irreducible components appearing in an irreducible decomposition of these ideals. Moreover, we give an explicit formula for the saturation number of monomial ideals in two variables.

Let $ e_1, ..., e_c $ be positive integers and let $ Y \subseteq \mathbb{P}^n$ be the monomial complete intersection defined by the vanishing of $x_1^{e_1}, ..., x_c^{e_c}$. In this paper we study sharp upper bounds on the number of equations and syzygies of subschemes parametrized by the Hilbert scheme of points $Hilb^d(Y)$, and discuss applications to the Hilbert scheme of points $Hilb^d(X)$ of arbitrary complete intersections $X \subseteq \mathbb{P}^n$., Final version

We study exceptional minuscule Schubert varieties and provide the defining equations of the defining ideals of their intersection with the big open cell. We also provide the resolutions of these ideals and characterize some of them in terms of fundamental examples of ideals in the theory of Gorenstein ideals., Comment: 21 pages, comments welcome

We investigate the Castelnuovo-Mumford regularity and the multiplicity of the toric ring associated with a three-dimensional Ferrers diagram. In particular, in the rectangular case, we provide direct formulas for these two important invariants. Then, we compare these invariants for an accompanying pair of Ferrers diagrams under some mild conditions and bound the Castelnuovo-Mumford regularity for more general cases., Comment: Revised following the reviewer's suggestions. 27 pages and 6 figures. Comments are welcome

Let $(R,\mathfrak{m}, k)$ be a strictly local normal $k$-domain of positive characteristic and $P$ be a prime divisor on $X=\text{Spec } R$. We study the Galois category of finite covers over $X$ that are at worst tamely ramified over $P$ in the sense of Grothendieck--Murre. Assuming that $(X,P)$ is a purely $F$-regular pair, our main result is that every Galois cover $f \: Y \to X$ in that Galois category satisfies that $\bigl(f^{-1}(P)\bigr)_{\text{red}}$ is a prime divisor. We shall explain why this should be thought as a (partial) generalization of a classical theorem due to S.S.~Abhyankar regarding the \'etale-local structure of tamely ramified covers between normal schemes with respect to a divisor with normal crossings. Additionally, we investigate the formal consequences this result has on the structure of the fundamental group representing the Galois category. We also obtain a characteristic zero analog by reduction to positive characteristics following Bhatt--Gabber--Olsson's methods., Comment: 45 pages, comments are welcome, typos fixed, shortened down, some parts were rewritten to improve exposition, Proposition 4.8 was removed as it was flawed v3: Major revision, to apper in ANT

In this paper we examine some natural ideal conditions and show how graphs can be defined that give a visualization of these conditions. We examine the interplay between the multiplicative ideal theory and the graph theoretic structure of the associated graph. Behavior of these graphs under standard ring extensions are studied, and in conjunction with the theory, some classical results and connections are made., Comment: 24 pages

We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and $1432$-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier. We apply our formulas to compute Castelnuovo-Mumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of certain Kazhdan-Lusztig ideals, including those coming from open patches of Schubert varieties in Grassmannians. This provides a correction to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri (2015)., 26 pages, revised version corrects compilation errors affecting Examples 3.1, 3.3, and 7.1

We characterize Hilbert polynomials that give rise to Hilbert schemes with two Borel-fixed points and determine when the associated Hilbert schemes or their irreducible components are smooth. In particular, we show that the Hilbert scheme is reduced and has at most two irreducible components. By describing the singularities in a neighbourhood of the Borel-fixed points, we prove that the irreducible components are Cohen-Macaulay and normal. We end by giving many examples of Hilbert schemes with three Borel-fixed points., Comment: To appear in Journal of Algebra

Over a $d$-dimensional Cohen-Macaulay local ring admitting a canonical module the definable closure of the class of balanced big Cohen-Macaulay modules is $d$-cotilting and is the smallest such class containing the maximal Cohen-Macaulay modules. We describe its cotilting structure and contrast it to the largest $d$-cotilting class containing the maximal Cohen-Macaulay modules. The enables an implicit classification of all $d$-cotilting classes containing the maximal Cohen-Macaulay modules., Comment: 13 pages, typos corrected, slightly expanded

We show that the hessian map of quartic plane curves is a birational morphism onto its image, thus bringing new evidence for a very interesting conjecture of Ciro Ciliberto and Giorgio Ottaviani. Our new approach also yields a simpler proof of the similar property for cubic surfaces, which is already known by the work of these two authors.

Let $C:f=0$ be a reduced curve in the complex projective plane. The minimal degree $mdr(f)$ of a Jacobian syzygy for $f$, which is the same as the minimal degree of a derivation killing $f$, is an important invariant of the curve $C$, for instance it can be used to determined whether $C$ is free or nearly free. In this note we study the relations of this invariant $mdr(f)$ with a decomposition of $C$ as a union of two curves $C_1$ and $C_2$, without common irreducible components. When all the singularities that occur are quasihomogeneous, a result by Schenck, Terao and Yoshinaga yields finer information on this invariant in this setting. Using this, we give some geometrical criteria, the first ones of this type in the existing literature as far as we know, for a line to be a jumping line for the rank 2 vector bundle of logarithmic vector fields along a reduced curve $C$., Comment: v4. Proposition 5.5 modified, one reference updated, the rest unchanged

A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. Under certain mild conditions on a positive algebraic number $\alpha$, the additive monoid $M_\alpha$ of the evaluation semiring $\mathbb{N}_0[\alpha]$ is atomic. The atomic structure of both the additive and the multiplicative monoids of $\mathbb{N}_0[\alpha]$ has been the subject of several recent papers. Here we focus on the monoids $M_\alpha$, and we study its omega-primality and elasticity, aiming to better understand some fundamental questions about their atomic decompositions. We prove that when $\alpha$ is less than 1, the atoms of $M_\alpha$ are as far from being prime as they can possibly be. Then we establish some results about the elasticity of $M_\alpha$, including that when $\alpha$ is rational, the elasticity of $M_\alpha$ is full (this was previously conjectured by S. T. Chapman, F. Gotti, and M. Gotti)., 14 pages

Given a generically finite local extension of valuation rings $V \subset W$, the question of whether $W$ is the localization of a finitely generated $V$-algebra is significant for approaches to the problem of local uniformization of valuations using ramification theory. Hagen Knaf proposed a characterization of when $W$ is essentially of finite type over $V$ in terms of classical invariants of the extension of associated valuations. Knaf's conjecture has been verified in important special cases by Cutkosky and Novacoski using local uniformization of Abhyankar valuations and resolution of singularities of excellent surfaces in arbitrary characteristic, and by Cutkosky for valuation rings of function fields of characteristic $0$ using embedded resolution of singularities. In this paper we prove Knaf's conjecture in full generality., Comments very welcome!

A set of polynomials $M$ is called a {\it submodule} of $\mathbb{C} [x_1, \dots, x_n ]$ if $M$ is a translation invariant linear subspace of $\mathbb{C} [x_1, \dots, x_n ]$. We present a description of the submodules of $\mathbb{C} [x,y]$ in terms of a special type of submodules. We say that the submodule $M$ of $\mathbb{C} [x,y]$ is an {\it L-module of order} $s$ if, whenever $F(x,y)=\sum_{n=0}^N f_n (x) \cdot y^n \in M$ is such that $f_0 =\ldots = f_{s-1}=0$, then $F=0$. We show that the proper submodules of $\mathbb{C} [x,y]$ are the sums $M_d +M$, where $M_d =\{ F\in \mathbb{C} [x,y] \colon \textit{deg}_x F, Comment: 22 pages

In this paper we present a simple technique to derive certificates of non-realizability for a combinatorial polytope. Our approach uses a variant of the classical algebraic certificates introduced by Bokowski and Sturmfels (1989), the final polynomials. More specifically we reduce the problem of finding a realization to that of finding a positive point in a variety and try to find a polynomial with positive coefficients in the generating ideal (a positive polynomial), showing that such point does not exist. Many, if not most, of the techniques for proving non-realizability developed in the last three decades can be seen as following this framework, using more or less elaborate ways of constructing such positive polynomials. Our proposal is more straightforward as we simply use linear programming to exhaustively search for such positive polynomials in the ideal restricted to some linear subspace. Somewhat surprisingly, this elementary strategy yields results that are competitive with more elaborate alternatives, and allows us to derive new examples of non-realizable combinatorial polytopes Gouveia was partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020 , funded by the Portuguese Government through FCT/MCTES . Macchia was supported by the Einstein Foundation Berlin under Francisco Santos grant EVF-2015-230 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project number 454595616 . Wiebe was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) [ PDF - 557980 - 2021 ], and by the Pacific Institute for the Mathematical Sciences (PIMS). The research and findings may not reflect those of the Institute.

The asymptotic Samuel function generalizes to arbitrary rings the usual order function of a regular local ring. Here we explore some natural properties in the context of excellent, equidimensional rings containing a field. In addition, we establish some results regarding the Samuel Slope of a local ring. This is an invariant related with algorithmic resolution of singularities of algebraic varieties. Among other results, we study its behavior after certain faithfully flat extensions., 21 pages

We prove a Galois-type correspondence between compositions of purely inseparable field extensions (including infinite ones) and subalgebras of differential operators. This correspondence can be utilized to establish a connection between separable field extensions and purely inseparable field extensions. Specifically, it serves as a progression towards gaining a deeper comprehension of a foliation theory on varieties in positive characteristic., 29 pages. Comments welcome!

In this note, we investigate the Baer splitting problem over commutative rings. In particular, we show that if a commutative ring $R$ is $\tau_q$-semisimple, then every Baer $R$-module is projective.

A ring is *unit-additive* if a sum of units is always either a unit or nilpotent. Examples, nonexamples, and constructions of such rings are explored here. The related concepts of unit-additivity dimension and unit-additive closure are also explored. Connections are made with Jacobson radicals, algebraic geometry, field theory, and Krull dimension., 16 pages. Comments welcome!

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$, $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we characterize all commutative Artinian rings whose prime ideal sum graphs are line graphs. Finally, we give a description of all commutative Artinian rings whose prime ideal sum graph is the complement of a line graph., 7 Figures

We survey some recent development in the stability theory of klt singularities. The main focus is on the solution of the stable degeneration conjecture., Submission to Handbook of Singularities. 42 pages. Comments are welcome

Let $R$ be a ring and $S$ a multiplicative subset of $R$. We introduce and study the notions of ($u$-)$S$-$w$-Noetherian modules and ($u$-)$S$-$w$-principal ideal modules. Some characterizations of these new concepts are given. In particular, we positively answer two open questions proposed by Kim and Lim [12, Question 14] and Lim [8, Question 8]., arXiv admin note: substantial text overlap with arXiv:2201.07913 by other authors

Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective space, we are able to generalize the necessary tools to deal with this case as well: we obtain a universal Gr\"obner basis for the vanishing ideal of the set of standard representatives of the projective space and we are able to reduce any monomial with respect to this Gr\"obner basis. With respect to the parameters of these codes, by considering subfield subcodes of projective Reed-Muller codes we are able to obtain long linear codes with good parameters over a small finite field.

Given an ideal $I$ in a polynomial ring $K[x_1,\dots,x_n]$ over a field $K$, we present a complete algorithm to compute the binomial part of $I$, i.e., the subideal ${\rm Bin}(I)$ of $I$ generated by all monomials and binomials in $I$. This is achieved step-by-step. First we collect and extend several algorithms for computing exponent lattices in different kinds of fields. Then we generalize them to compute exponent lattices of units in 0-dimensional $K$-algebras, where we have to generalize the computation of the separable part of an algebra to non-perfect fields in characteristic $p$. Next we examine the computation of unit lattices in affine $K$-algebras, as well as their associated characters and lattice ideals. This allows us to calculate ${\rm Bin}(I)$ when $I$ is saturated with respect to the indeterminates by reducing the task to the 0-dimensional case. Finally, we treat the computation of ${\rm Bin}(I)$ for general ideals by computing their cellular decomposition and dealing with finitely many special ideals called $(s,t)$-binomial parts. All algorithms have been implemented in SageMath., 29 pages

We review the open problems in the theory of deformations of zero-dimensional objects, such as algebras, modules or tensors. We list both the well-known ones and some new ones that emerge from applications. In view of many advances in recent years, we can hope that all of them are in the range of current methods.

In his paper and thesis in 1989, Ziegler posed several conjectures regarding commutative algebra related to hyperplane arrangements. In this article, we revisit two of them. One is on generic cuts of free arrangements, and the other has to do with minimal degree generators for the logarithmic differential forms. We prove the first one, and disprove the second one. We also give some positive answers to related problems he posed, using recent developments in arrangement theory., comments welcome!

$\DeclareMathOperator{\IntR}{Int{}^\text{R}}$$\DeclareMathOperator{\Int}{Int}$Let $D$ be a domain. Park determined the necessary and sufficient conditions for which the ring of integer-valued polynomials $\Int(D)$ is a globalized pseudovaluation domain (GPVD). In this work, we investigate the ring of integer-valued rational functions $\IntR(D)$. Since it is necessary that $D$ be a GPVD for $\IntR(D)$ to be a GPVD, we consider $\IntR(D)$, where $D$ is a GPVD. We determine that if $D$ is a pseudosingular GPVD, then $\IntR(D)$ is a GPVD. We also completely characterize when $\IntR(D)$ is a GPVD if $D$ is a pseudovaluation domain that is not a valuation domain.

Let $(R, \mathfrak{m})$ be a $d$-dimensional Noetherian local ring that is formally equidimensional, and let $M$ be an arbitrary $R$-submodule of the free module $F = R^p$ with an analytic spread $s:=s(M)$. In this work, inspired by Herzog-Puthenpurakal-Verma in \cite{herzog}, we show the existence of an unique largest $R$-module $M_{k}$ with $\ell_R(M_{k}/M)

Macaulay posets are posets in which an analog of the Kruskal-Katona Theorem holds. Macaulay rings (also called Macaulay-Lex rings) are rings in which an analog of Macaulay's Theorem for lex ideals holds. The study of both of these objects started with Macaulay almost a century ago. Since then, these two branches have developed separately over the past century, with the last link being the Clements-Lindstr\"om Theorem. For every ring that is the quotient of a polynomial ring by a homogeneous ideal we define the poset of monomials. Under certain conditions, we prove a Macaulay Correspondence Theorem, a ring is Macaulay if and only if its poset of monomials is Macaulay. Furthermore, the tensor product of rings corresponds to the Cartesian product of the posets of monomials. This allows us to transfer results between rings and posets. By using this translation, we give several answers to a problem posed by Mermin and Peeva, a positive answer to Hoefel's question about applying Macaulay poset theory to ring theory, and deduce several other results in algebra and combinatorics. A new proof of the Mermin-Murai Theorem on colored square free rings is presented by using star posets. We extend the Mermin-Murai Theorem to rings that are not square free. Using a result from Mermin and Peeva we give an answer to a question posed by Bezrukov and Leck. Some results of Chong also give answers to the Bezrukov-Leck problem. All of these results have a common feature. They involve the tensor product of rings whose Hasse graphs of the poset of monomials are trees. We call such rings, tree rings. We give a classification of Macaulay rings that are the tensor product of a tree ring. Finally, we show that there are Macaulay rings that are not the tensor product of tree rings, and present the first examples of Macaulay rings that are not quotients by a monomial ideal and not quotients by a toric ideal.

Let G be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of G which depends only on the width of G, that is, the difference between the largest and the smallest generators of G. In this way, we make progress towards a conjecture of Herzog and Stamate. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width.

Let $k$ be an infinite field of characteristic $p > 0$ and let $R = k[Y_1,\ldots, Y_d]$ (or $R = k[[Y_1,\ldots, Y_d]]$). Let $F \colon \text{Mod}(R) \rightarrow \text{Mod}(R)$ be the Frobenius functor and let $\mathcal{M}$ be a $F_R$-finite module (in the sense of Lyubeznik \cite{Lyu-2}). We show that if $r \geq 1$ then the Koszul homology modules $H_i(Y_1,\ldots, Y_r; \mathcal{M})$ are $F_{\overline{R}}$-finite modules where $\overline{R} = R/(Y_1,\ldots, Y_r)$ for $i = 0, \ldots, r$. As an application if $A$ is a regular ring containing a field of characteristic $p > 0$ and $S = A[X_1,\ldots, X_m]$ is standard graded and $I$ is an arbitrary graded ideal in $S$ then we give a comprehensive study of graded components of local cohomology modules $H^i_I(S)$. This extends in positive characteristic results we proved in \cite{P}. We study $H^i_I(S) $ when $A$ is local and prove that if $S/I$ is equidimensional and $Proj(S/I)$ is Cohen-Macaulay then $H^i_I(S)_n = 0$ for all $n \geq 0$ and for all $i > \ height \ I$. If $B$ is a equicharacteristic local Noetherian ring with infinite residue field and with a surjective map $π\colon T \rightarrow B$ where $(T,\mathfrak{n})$ is regular local then we show that the Koszul cohomology modules $H^j(\mathfrak{n}, H^{\dim T - i}_{\ker π}(T))$ depend only on $A, i, j$ and not on $T$ and $π$., arXiv admin note: text overlap with arXiv:1701.01270

Let $H = \langle n_1, n_2, n_3\rangle$ be a numerical semigroup. Let $\tilde H$ be the interval completion of $H$, namely the semigroup generated by the interval $\langle n_1, n_1+1, \ldots, n_3\rangle$. Let $K$ be a field and $K[H]$ the semigroup ring generated by $H$. Let $I_H^*$ be the defining ideal of the tangent cone of $K[H]$. In this paper, we describe the defining equations of $I_H^*$. From that, we establish the Herzog-Stamate conjecture for monomial space curves stating that $\beta_i(I_H^*) \le \beta_i(I_{\tilde H}^*)$ for all $i$, where $\beta_i(I_H^*)$ and $\beta_i(I_{\tilde H}^*)$ are the $i$th Betti numbers of $I_H^*$ and $I_{\tilde H}^*$ respectively.

We investigate the injectivity of the Frobenius map on thickenings of smooth varieties in projective space over a field of positive characteristic. We obtain uniform bounds -- i.e., independent of the characteristic -- on the thickening that ensures an injective Frobenius map when the projective variety is a smooth complete intersection or an arbitrary projective embedding of an elliptic curve. Our bounds are sharp in the case of hypersurfaces, and in the case of elliptic curves., Comments welcome!

Let $A$ be a commutative Noetherian ring of characteristic zero and $R=A[X_1, \ldots, X_d]$ be a polynomial ring over $A$ with the standard $\mathbb{N}^d$-grading. Let $I\subseteq R$ be an ideal which can be generated by elements of the form $aU$ where $a \in A$ (possibly nonunit) and $U$ is a monomial in $X_i$'s. We call such an ideal as a `$\mathfrak{C}$-monomial ideal'. Local cohomology modules supported on monomial ideals gain a great deal of interest due to their applications in the context of toric varieties. It was observed that for $\underline{u} \in \mathbb{Z}^d$, their $\underline{u}^{th}$ components depend only on which coordinates of $\underline{u}$ are negative. In this article, we show that this statement holds true in our general setting, even for certain invariants of the components. We mainly focus on the Bass numbers, injective dimensions, dimensions, associated primes, Bernstein-type dimensions, and multiplicities of the components. Under the extra assumption that $A$ is regular, we describe the finiteness of Bass numbers of each component and bound its injective dimension by the dimension of its support. Finally, we present a structure theorem for the components when $A$ is the ring of formal power series in one variable over a characteristic zero field.

Using discrete Morse theory, Batzies and Welker introduced Morse resolutions of monomial ideals. In this note, we present the {\it Macaulay2} package {\tt MorseResolutions} for working with two important classes of Morse resolutions: Lyubeznik and Barile-Macchia resolutions. This package also contains procedures to search for a minimal Barile-Macchia resolution of a given monomial ideal., 9 pages

The purpose of this paper is to prove that one can read off the gonality sequence of a smooth projective curve from syzygies of secant varieties of the curve embedded by a line bundle of sufficiently large degree. More precisely, together with Ein-Niu-Park's theorem, our main result shows that the gonality sequence of a smooth projective curve completely determines the shape of the minimal free resolutions of secant varieties of the curve of sufficiently large degree. This is a natural generalization of the gonality conjecture on syzygies of smooth projective curves established by Ein-Lazarsfeld and Rathmann to the secant varieties., 22 pages, any comments are welcome

The set of nonnegative integer lattice points in a polytope, also known as the fiber of a linear map, makes an appearance in several applications including optimization and statistics. We address the problem of sampling from this set using three ingredients: an easy-to-compute lattice basis of the constraint matrix, a biased sampling algorithm with a Bayesian framework, and a step-wise selection method. The bias embedded in our algorithm updates sampler parameters to improve fiber discovery rate at each step chosen from previously discovered elements. We showcase the performance of the algorithm on several examples, including fibers that are out of reach for the state-of-the-art Markov bases samplers., 22 pages, 12 figures