CyberSec Research

Linear resolutions and the stronger notion of linear quotients are important properties of monomial ideals. In this paper, we fully characterize linear resolutions and linear quotients in terms of the lcm-lattice of monomial ideals. These results complement characterizations of these two properties in terms of the Alexander dual of the corresponding Stanley-Reisner simplicial complex. In addition, we discuss applications to the case of edge ideals.

We study the number $O_d$ of finite $O$-sequences of a given multiplicity $d$, with particular attention to the computation of $O_d$. We show that the sequence $(O_d)_d$ is sub-Fibonacci, and that if the sequence $(O_d / O_{d-1})_d$ converges, its limit is bounded above by the golden ratio. This analysis also produces an elementary method for computing $O_d$. In addition, we derive an iterative formula for $O_d$ by exploiting a decomposition of lex-segment ideals introduced by S. Linusson in a previous work.

In his work on the Bass series of syzygy modules of modules over a commutative noetherian local ring $R$, Lescot introduces a numerical invariant, denoted $\sigma(R)$, and asks whether it is finite for any $R$. He proves that this is so when $R$ is Gorenstein or Golod. In the present work many new classes of rings $R$ for which $\sigma(R)$ is finite are identified. The new insight is that $\sigma(R)$ is related to the natural map from the usual cohomology of the module to its stable cohomology, which permits the use of multiplicative structures to study the question of finiteness of $\sigma(R)$.

We compute the maximum number of rational points at which a homogeneous polynomial can vanish on a weighted projective space over a finite field, provided that the first weight is equal to one. This solves a conjecture by Aubry, Castryck, Ghorpade, Lachaud, O'Sullivan and Ram, which stated that a Serre-like bound holds with equality for weighted projective spaces when the first weight is one, and when considering polynomials whose degree is divisible by the least common multiple of the weights. We refine this conjecture by lifting the restriction on the degree and we prove it using footprint techniques, Delorme's reduction and Serre's classical bound.

We translate the operations of polarization and depolarization from monomial ideals in a polynomial ring to abstract simplicial complexes. As a result, we explicitly describe the relation between the Koszul simplicial complex of a monomial ideal and that of its polarization. Using the simplicial translation of depolarization we propose a way to reduce a simplicial complex to a smaller one with the same homology. This type of reduction, that can be interpreted as non-elementary collapse, can be used as a pre-process step for algorithms on simplicial complexes. We apply this methodology to the efficient computation of the Alexander dual of abstract simplicial complexes.

In this paper we study the structure of the coordinate ring of an affine ind-variety. We prove that any coordinate ring of an affine ind-variety which is not isomorphic to an affine algebraic variety does not have a countable set of generators. Also we prove that coordinate rings of affine ind-varieties have an everywhere dense subspace of countable dimension.

Let $(R,\mathfrak{m})$ be a Noetherian local ring, and let $J$ be an arbitrary ideal of $R$. Suppose $M$ is a finitely generated $R$-module. Let $x_1,\ldots,x_r$ be a $J$-filter regular sequence on $M$. We provide an explicit number $N$ such that the Bass and Betti numbers of $M/(x_1, \ldots, x_r)M$ are preserved when we perturb the sequence $x_1, \ldots,x_r$ by $\varepsilon_1, \ldots, \varepsilon_r \in \mathfrak{m}^N$.

We provide explicit formulas for key invariants of special fiber rings of ladder determinantal modules, that is, modules that are direct sums of ideals of maximal minors of a ladder matrix. Our results are given in terms of the combinatorial data of the associated ladder matrix. In particular, we compute its dimension, regularity, $a$-invariant, and multiplicity, which via \textsc{Sagbi} degeneration coincide with those of Hibi rings associated to a distributive lattice. Then, via Gr\"{o}bner degeneration these calculations are reduced to quotients of polynomial rings by monomial ideals. Our formula for the multiplicity of the special fiber ring of these ladder determinantal modules is obtained by counting the number of standard skew Young tableaux associated to a certain skew partition, and so provides a natural generalization of the classical formula for the degree of the Grassmannian.

Let $G=\mathrm{GL}_n(\mathbb{F})$, $\mathrm{O}_n(\mathbb{F})$, or $\mathrm{Sp}_{2n}(\mathbb{F})$ be one of the classical groups over an algebraically closed field $\mathbb{F}$ of characteristic $0$, let $\breve{G}$ be the MVW-extension of $G$, and let $\mathfrak{g}$ be the Lie algebra of $G$. In this paper, we classify the closed orbits in the enhanced standard representation $\mathfrak{g}\times E$ of $G$, where $E$ is the natural representation if $G=\mathrm{O}_n(\mathbb{F})$ or $\mathrm{Sp}_{2n}(\mathbb{F})$, and is the direct sum of the natural representation and its dual if $G=\mathrm{GL}_n(\mathbb{F})$. Additionally, for every closed $G$-orbit in $\mathfrak{g}\times E$, we prove that it is $\breve{G}$-stable, and determine explicitly the corresponding stabilizer group as well as the action on the normal space.

We recall a higher dimension analog of the classic plane de Jonqui\`eres transformations, as given by Hassanzadeh and Simis. Such a parameterization defines a birational map from $\mathbb{P}^{n-1}$ to a hypersurface in $\mathbb{P}^{n}$, and a natural question that arises is how to obtain its implicit equation. We pass from the image of this map to its graph, and implicitize the Rees algebra of the ideal of the de Jonqui\`eres map when its underlying Cremona support is tame. We then consider the Rees rings of ideals of generalized de Jonqui\`eres transformations, and answer a conjecture of Ramos and Simis.

To a generic hypersurface in the affine torus $(\mathbb{C}^*)^n$ we associate a hypersurface arrangement in the projective space $\mathbb{P}^n$ consisting of the $n+1$ coordinate hyperplanes and a generic hypersurface, and compute the minimal graded resolutions of the corresponding Jacobian algebra.

Given a field $\mathbb{F}$, we introduce a novel group $SD(\mathbb{F})$ of its self-maps: the solutions $f \colon \mathbb{F} \twoheadrightarrow \mathbb{F}$ to the functional equation $f \left( (x+y)/(x-y) \right) = (f(x) + f(y))/(f(x) - f(y))$ for all $ x \neq y$ in $\mathbb{F}$. We compute this group for all fields algebraic over $\mathbb{F}_p$. In particular, this group distinguishes $\mathbb{F}_5$ among all finite fields $\mathbb{F}_q$, and in fact among all subfields of $\overline{\mathbb{F}_q}$.

We consider the relationship between the Stanley-Reisner ring (a.k.a. face ring) of a simplicial or boolean complex $\Delta$ and that of its barycentric subdivision. These rings share a distinguished parameter subring. S. Murai asked if they are isomorphic, equivariantly with respect to the automorphism group $\operatorname{Aut}(\Delta)$, as modules over this parameter subring. We show that, in general, the answer is no, but for Cohen-Macaulay complexes in characteristic coprime to $|\operatorname{Aut}(\Delta)|$, it is yes, and we give an explicit construction of an isomorphism. To give this construction, we adapt and generalize a pair of tools introduced by A. Garsia in 1980. The first one transfers bases from a Stanley-Reisner ring to closely related rings of which it is a Gr\"obner degeneration, and the second identifies bases to transfer.

The solution to the general univariate polynomial equation has been sought for centuries. It is well known there is no general solution in radicals for degrees five and above. The hyper-Catalan numbers $C[m_2,m_3,m_4,\ldots]$ count the ways to subdivide a planar polygon into exactly $m_2$ triangles, $m_3$ quadrilaterals, $m_4$ pentagons, etc. Wildberger and Rubine (2025) show the generating series $\mathbf{S}$ of the hyper-Catalan numbers is a formal series zero of the general geometric polynomial (meaning, general except for a constant of $1$ and a linear coefficient of $-1$). Using a variant of the series solution to the geometric polynomial that has the number of vertices, edges, and faces explicitly shown, We prove their infinite series result may be viewed as a finite identity at each level, where a level is a truncation of $\mathbf{S}$ to a given maximum number of vertices, edges, or faces (bounded by degree). We illustrate this result, as well as the general correspondence between operations on sets of subdivided polygons and the algebra of polynomials, with figures and animations generated using Python.

In 2016 Ananyan and Hochster proved Stillman's conjecture by showing the existence of a uniform upper bound on the lengths of an $R_\eta$-sequence containing a fixed $n$ forms of degree at most $d$ in polynomial rings over a field. This result yields many other uniform bounds including bounds on the projective dimension of the ideals generated by $n$-forms of degree at most $d$. Explicit values of these bounds for forms of degrees $5$ and higher are not yet known. This article constructs such explicit bounds, one of which is an upper bound for the projective dimension of all homogeneous ideals, in polynomial rings over a field, generated by $n$ forms of degree at most $d$. In the settings of the Eisenbud-Goto conjecture, we derive an explicit bound of the Castelnuovo-Mumford regularity of a degenerate prime ideal $P$ in a polynomial ring $S$ in terms of the multiplicity of $S/P$.

We introduce characteristic numbers of a finite commutative unital $\mathbb{C}$-algebra, which are numerical invariants arising from algebraic intersection theory. We characterize Gorenstein and local complete intersection algebras in terms of their characteristic numbers. We compute characteristic numbers for certain families of algebras. We show that characteristic numbers are constant on $\mathrm{Hilb}_d(\mathbb{A}^1)$, provide an explicit upper bound for characteristic numbers on the smoothable component of $\mathrm{Hilb}_d(\mathbb{A}^n)$ and an explicit lower bound for characteristic numbers on the Gorenstein locus of $\mathrm{Hilb}_d(\mathbb{A}^n)$ for $n \geq d-2$.

Let $\varphi: R\rightarrow S$ be a flat local homomorphism between commutative Noetherian local rings. In this paper, the ascent and descent of Artinian module structures between $R$ and $S$ are investigated. For an Artinian $R$-module $A$, the structure of $A\otimes_RS$ is described. As an application, the Artinianess of certain local cohomology modules is clarified.

For a field $\mathbb{F}$, what are all functions $f \colon \mathbb{F} \rightarrow \mathbb{F}$ that satisfy the functional equation $f \left( (x+y)/(x-y) \right) = (f(x) + f(y))/(f(x) - f(y))$ for all $ x \neq y$ in $\mathbb{F}$? We solve this problem for the fields $\mathbb{Q}, \mathbb{R}$, and a class of its subfields that includes the real constructible numbers, the real algebraic numbers, and all quadratic number fields. We also solve it over the complex numbers and on any subfield of $\mathbb{R}$, if $f$ is continuous over the reals. The proofs involve a mix of algebra in all fields, analysis over the real line, and some topology in the complex plane.