Papers

We study numerical invariants of homogeneous ideals generated by three forms, with particular emphasis on the interaction between the initial degree of the syzygy module, the multiplicity of the codimension-two part of the ideal, and the Bourbaki degree. We extend the Bourbaki degree from gradient ideals of plane curves to arbitrary three-generated homogeneous ideals and obtain sharp numerical bounds in both the general and the equigenerated settings. In the equigenerated case, these bounds extend the du Plessis–Wall picture and lead to the study of numerical, structural, dimensional, and integrable gaps. We show that numerical admissibility does not in general imply realizability and obtain further restrictions from the minimal graded free resolution. As an application, we determine the possible Bourbaki degrees of reduced singular quartic plane curves and describe them in terms of their singularity configurations; the possible values are 0,1,2,3,4,6,7,8.

We provide first examples of border rank lower bounds obtained using border apolarity beyond “weak border apolarity”. Border apolarity introduced by Buczyńska-Buczyński characterizes tensors of border rank greater than $r$ as those whose annihilator does not contain an ideal with certain properties depending on $r$ and the ambient space of the tensor. Previous applications of border apolarity depended on the algorithm developed by Conner-Harper-Landsberg. For a tensor with a large symmetry group it lists a set of ideals that satisfy some of the properties mentioned above. We give first examples where the list is non-empty but we are able to prove that no ideal on this list satisfies all necessary conditions coming from border apolarity. The new lower bound examples include tensors of interest in complexity: the first Schönhage triple sum of matrix multiplication tensors where the border rank was unknown, and the direct sum of two copies of a tensor that could potentially be used to prove the exponent of matrix multiplication is two. A third example of interest in group theory related to work of A. Leitner is also analyzed.

Let $C_n$ be the cycle of length $n\ge3$ and let $I_{C_n}$ be its cut ideal. We show that $I_{C_n}$ has a quadratic Gröbner basis with respect to an explicit weight order. Since the defining configuration consists of $(0,1)$-vectors, the initial monomials of such a basis are automatically squarefree. As the cut polytope of a cycle is the parity polytope, the result gives a regular unimodular flag triangulation of this classical polytope. Together with the known tree case and the clique-sum theorem for cut ideals, the cycle result also yields a quadratic Gröbner basis for the cut ideal of every connected ring graph with at least one edge, thereby supplying the missing cycle input and establishing the result for connected ring graphs.

We introduce and study the omega invariant of a proper ideal in a Noetherian commutative ring, defined as the number of associated primes of the ideal. Our main objective is to investigate this invariant for monomial ideals and their powers. We characterize associated primes through monomial witnesses and provide an algorithmic procedure for constructing such witnesses from the exponent vectors of the minimal generators. These results lead to explicit formulas and bounds for the omega invariant without requiring the computation of a primary decomposition. We further establish alternative descriptions using irreducible decompositions and Alexander duality. A matrix-based approach is developed to detect associated primes of powers of monomial ideals directly from the exponent matrix of the original ideal. We also investigate the behavior of witnesses under passage from $I^n$ to $I^{n+1}$ and derive corresponding results for edge ideals of graphs.

Recently, Jankowski introduced the Serre criteria for superschemes to define the super versions of normality and reducedness (called generic fermionic regularity and reducedness). We show that for a flat, proper morphism $f:X\rightarrow Y$, where $X$ is a superscheme and $Y$ is a Dedekind scheme of finite type over a field of characteristic $0$, the set of $y\in Y$ for which the fibers $X_y$ are normal, GFRR, regular, and Cohen-Macaulay is open in $Y$. In particular, if one has a degeneration over $\mathbb{A}_{\mathbb{C}}^{1|0}$, this shows that these properties can be lifted from the special fiber to the generic one.

Let $S=\kk[x_1,\ldots,x_n]$ with its standard $\mathbb N^n$-grading. For $0\le i<n$, we identify the finite-box local-cohomology category $\mathcal H_i(\mathbf t)$ with the representation category of the upper support poset $U_{n-i}(\mathbf t)$. The local-cohomology functor admits an exact functorial section selecting sources of minimal projective dimension and maximal depth in every non-top degree, but for $n\ge2$ and $\mathbf t\ge\mathbf1$ no fully faithful section exists for $i=1$ or $i=n-1$, even without exactness. We construct explicit sources in $\mathcal A^+_{\mathbf t}$ for all indecomposable projective objects. The source attached to $\mathbf a$ is a prescribed syzygy of a Brun–Fløystad interval module; it is a brick of projective dimension $n-i$ and is Cohen–Macaulay exactly on the bottom support-rank stratum $\ell(\mathbf a)=n-i$. We relate these sources in adjacent degrees to the support-rank torsion filtration, obtaining a syzygy tower, and identify their additive category with the projective part of $\mathcal H_i(\mathbf t)$. As applications, we determine attached primes from projective covers and compute the Bass numbers of projective and injective objects. We also pose questions about extensions to complete-intersection and general monomial supports.

This paper deals with certain classes of modules under module-finite extensions. Let $\varphi: R\hookrightarrow S$ be a module-finite extension between commutative Noetherian local rings. We investigate the transfer of Artinian module structures and attached primes between $R$ and $S$. We clarify the behavior of local cohomology modules as well as certain structures of finitely generated $S$-modules under the restriction of scalars to $R$ via $\varphi$. We show that $R$ is a quotient of a Cohen-Macaulay local ring if and only if so is $S$. As an application, we characterize the structure of Nagata's idealization. Using Macaulayfication of algebraic varieties and idealization, we give an example to illustrate the results.

It has been asked whether a reduced non-McCoy ring can satisfy both the annihilator condition and $(A_n)$ for some $n \geq 2$. In this work, we prove that such rings exist exactly when $n=2$, by constructing a countable example and showing that the annihilator condition together with $(A_3)$ forces a ring to be McCoy.

It was conjectured that every integral domain $D$ satisfies $\dim\operatorname{Int}(D) \leq \dim D[X]$. We disprove this conjecture and, more strongly, classify all pairs for which both dimensions are finite. Writing $m=\dim D[X]$ and $n=\dim\operatorname{Int}(D)$, the pairs that occur are precisely $\{(1,1),(2,2)\}\ \cup\ \{(m,n)\in\mathbb Z_{>0}^{2}:m\geq3,\ n\geq m-1\}$.

We describe a new greatest common divisor (GCD) algorithm for polynomials with integer coefficients. The bit complexity of the new algorithm is polynomial in the input and output sizes and the individual degree bounds.Our algorithm follows the standard approach by reducing multivariate polynomial GCD to univariate polynomial GCD. Our algorithm reduces a multivariate polynomial GCD to a single univariate polynomial GCD. The main idea of our algorithm is a new variable substitution which reduces a multivariate polynomial to a separated one, that is, the coefficients in a main variable are all monomials. The explicit bit complexity is analyzed and we have implemented our algorithm in Maple. It is shown that our algorithm is efficient for polynomials with high degree, large number of variables, but small number of terms in GCD.

We show the existence of counterexamples to a four-decade-old conjecture attributed to Stanley concerning the unimodality of $h^*$-polynomials of IDP polytopes. As additional applications of our main constructions, we also disprove a conjecture by Brenti on the log-concavity of $h^*$-polynomials of Gorenstein IDP polytopes, and a conjecture by Ferroni and Higashitani concerning the log-concavity of the Ehrhart series of IDP polytopes. We also answer their question about the existence of very ample polytopes with non-log-concave interior Ehrhart series. Our class of examples arises by taking Cayley sums of rectangular prisms, and hence they possess regular unimodular flag triangulations by a result of Haase, Paffenholz, Piechnik and Santos. For the unimodality conjecture, we can even find smooth counterexamples.

Let $(A,\mathfrak{m})$ be a Gorenstein local ring with $G(A)$ Cohen-Macaulay, and let $M$ be a Cohen-Macaulay $A$-module of finite projective dimension. In \cite{Quasipure}, the authors proved that $e_1(M)\geq \binom{c+1}{2}$, where $c=\operatorname{reg}G(A)$ and $e_i(M)$ is the $i$th Hilbert coefficient of $M$. We first show that this bound remains valid when $A$ is Cohen-Macaulay. We then study upper bounds for $e_2(M)$ when $e_1(M)=\binom{c+1}{2}+i$ for $i=1,2$, and investigate the consequences of equality. In particular, we obtain depth properties and explicit descriptions of the $h$-polynomial of $G(M)$. Finally, we extend these results to strict complete intersection rings without assuming that $M$ has finite projective dimension.

Let $\kk$ be a field of characteristic $p$ and $R=\kk[x_1,\dots,x_n]$. Let $D$ denote the ring of $\kk$-linear differential operators on $R$. Let $\varphi:R\to R$ be a $\kk$-algebra endomorphism whose Jacobian is a unit in $R$. We show that there is a unique $\kk$-algebra endomorphism $\Phi:D\to D$ such that $\Phi|_R=\varphi$ and that the restriction of scalar via $\Phi$ preserves holonomicity. We also construct an example of a $\kk$-algebra endomorphism of $D$ that does not preserve holonomicity.

We use the full multiplication formula to study syzygy and suspension of cluster characters. In stably 2-Calabi–Yau Frobenius categories, syzygy and cosyzygy induce inverse automorphisms of localized character algebras; in Hom-finite 2-Calabi–Yau triangulated categories, the same uniqueness principle transports all cluster characters under suspension. The latter gives our Auslander–Reiten $F$-polynomial identity. No mutation reachability is required. We extend positroid twist identities to all objects, recover the Grassmannian and unipotent-cell formulas, identify partition-function algebras before boundary localization, and derive tropical covariance when the relevant tropical coordinate map is a homeomorphism.

We study reconstruction and profile geometry for divisor schemes of binary linear systems and their sparse Krylov charts. Over the integers, the first nonzero equations of the complete embedded divisor scheme recover the defining linear system functorially under arbitrary base change, yielding a closed immersion from the Grassmannian of linear systems to the corresponding Hilbert scheme. For monomial systems in characteristic zero, arithmetic profiles classify the reduced factorization branches, determine their image dimensions and generic multiplicities, and control geometric reducedness. For complete progressions, the normalizations of the relation branches and their images are explicit products of projective spaces equipped with two natural polarizations. We give an affine normality criterion in terms of the associated Fourier data and a projective criterion obtained by adjoining an endpoint-allocation condition. These results place the closed sparse Krylov rank loci in a uniform reconstruction–normalization framework and yield explicit mixed-degree and formal-profile consequences. They also clarify the limit of normalization data alone: recovering a possibly nonnormal image algebra requires additional information not addressed here.

We provide a combinatorial description of the stabilization index of the $v$-function of powers of edge ideals for arbitrary graphs.

Computation of the Frobenius closure of ideals in rings of prime characteristic is a difficult problem. For a given ideal, its Frobenius test exponent provides a valuable degree of uniformity in performing the calculation. Hence, it is desirable to find uniform upper bounds on the Frobenius test exponent of ideals. Even in nice rings of low dimension, Brenner showed considering the collection of all ideals is generally hopeless, but for Cohen-Macaulay rings, Katzman-Sharp there are uniform upper bounds on the Frobenius test exponent for the class of parameter ideals. Subsequent efforts in controlling the Frobenius test exponents have typically involved studying the Frobenius action on local cohomology and the degree to which this action is nilpotent. The goal of this survey article is to examine the history of the Frobenius test exponent problem and its relationships to singularity types for local rings defined in terms of the Frobenius action on local cohomology. We also explore several generalizations of these prior results to a setting where less nilpotence in the Frobenius action is assumed.

We give a complete characterization of the weak Lefschetz property (WLP) for monomial complete intersections over a field of positive characteristic. Richard Stanley observed that in characteristic zero WLP, and in fact the strong Lefschetz property (SLP), holds for all degree sequences, as a consequence of the hard Lefschetz theorem, and the same result was explained by Junzo Watanabe using the representation theory of $\mathfrak{sl}_2$. In positive characteristic, many partial results are known, most notably the classification for constant degree sequences due to Brenner–Kaid and Kustin–Vraciu. Our approach is based on a cohomological and representation-theoretic interpretation of WLP. Combined with an analysis of cohomology characters, this leads to a complete numerical criterion for WLP, expressed by simple inequalities involving the $p$-adic digits of the exponents. We also give a new proof of the known classification of SLP using Renaud's algorithm for multiplication in the Green–Han–Monsky ring.

Monge matrices arise throughout combinatorial optimization and algorithm design; the Monge polytope $\mathcal{M}_{pq}$ is the set of $p \times q$ Monge matrices lying inside the standard simplex on the set of matrix coordinates. We find a Stanley decomposition of the associated affine semigroup, and use it to obtain a volume formula for $\mathcal{M}_{pq}$ expressed as a sum over "Z-avoiding" Delannoy paths in a $p \times q$ grid. An efficient dynamic-programming implementation of this formula computes the volume in dimensions far beyond the reach of general-purpose exact-volume algorithms (e.g., the volume of $\mathcal{M}_{20,20}$, which has dimension 399, is computed in a fraction of a second). As a corollary of our Stanley decomposition, we also obtain a combinatorial closed form for the Ehrhart series of $\mathcal{M}_{pq}$.

We develop a general first-order theory of Schur apolarity for the study of secant varieties of flag varieties in arbitrary homogeneous embeddings. Extending the classical apolarity–fat-point correspondence for Veronese varieties, we show that in the Schur setting the algebraic square of the apolar ideal need not coincide with the geometric double-point conditions. We introduce a geometric Schur square whose relevant component is the conormal space, yielding a Schur Dual Terracini Lemma. Our construction recovers classical apolarity in the symmetric case. A slot-by-slot Consistency Theorem realizes these intrinsic conditions as multigraded double points. As an application, we determine the dimensions of all secant varieties of $\operatorname{Fl}(1,2;V_n)$ embedded by $\mathcal{O}(1,1)$: the only defective cases are $\sigma_2(\operatorname{Fl}(1,2;V_3))$ and $\sigma_3(\operatorname{Fl}(1,2;V_4))$, both of defect one.

Let $A, B \in \mathbb{Z}[x_1, \dots, x_n]$ be multivariate polynomials with integer coefficients and let $G = \gcd(A, B)$. We present an algorithm for computing $G$ whose expected bit complexity is asymptotically linear in all fundamental parameters: the number of variables $n$, the term count $T = \max\{\|A\|_0, \|B\|_0, \|G\|_0\}$, the total degree $D$, and the logarithmic coefficient sizes $\log\Hi$ and $\log\Ho$, where $\Hi$ bounds the coefficients of the inputs and $\Ho$ bounds those of the GCD. The bit complexity is characterized by the clean bound \[ \widetilde{O}\bigl( n \cdot T \cdot D \cdot \log\Hi \cdot \log\Ho \bigr). \] To our knowledge, this is the first sparse GCD algorithm over the integers that achieves linear complexity in all these parameters simultaneously. The integer algorithm is built upon a new field GCD algorithm. For $A, B \in \K[x_1, \dots, x_n]$ over a field $\K$ with $\operatorname{char}(\K) = 0$ or $\operatorname{char}(\K) > \deg G$, we give the first algorithm that computes $G = \gcd(A,B)$ with expected \[ \widetilde{O}\bigl( n \cdot T \cdot D \bigr) \] field operations, which is both input- and output-sensitive. The key technical contribution behind both algorithms is a derivative-aided separated Hensel lifting technique introduced in this paper. By introducing an auxiliary variable and leveraging derivative information, our scheme extracts all partial exponents via a single $z^2$-lift per variable, achieving constant sequential depth $O(1)$. This stands in sharp contrast to classical Hensel lifting, which requires $O(D)$ sequential lifting steps and suffers from representation densification in the sparse setting. The field algorithm is then extended to the integer case through modular reduction and rational reconstruction.

Let $$ S=\mathbb{C}[x,y,z],\qquad p_m=x^m+y^m+z^m. $$ For every $r\geq1$, we prove that $$ p_r,\ p_{r+1},\ p_{3r+1} \qquad\text{and}\qquad p_r,\ p_{r+1},\ p_{3r+2} $$ are regular sequences exactly when $r\not\equiv1\pmod3$. This proves the Conca–Krattenthaler–Watanabe prediction for two infinite families. The proof is elementary. Four simple reductions modulo $(p_r,p_{r+1})$ reduce the problem to the two equations $x+y+z=0$ and $xy+xz+yz=0$.

We give Hodge-theoretic characterizations of characteristic-zero varieties of open $F$-nilpotent and open weakly $F$-nilpotent type, extending a result of Srinivas and Takagi beyond isolated singularities. We prove one direction unconditionally and the converse assuming the weak ordinarity conjecture.

Deciding whether a system of multivariate quadratic equations over $\mathbb F_2$ has a solution is a classical NP-complete problem, and remains so for square systems, with as many equations as variables. The hardness of this problem is one of the cornerstones of nowadays post-quantum cryptography. Let $\MQ_0(n)$ and $\MQ_1(n)$ denote the sets of square quadratic systems in $n$ variables having respectively no solutions and exactly one solution. $\cup_{n\geq 2} \MQ_0(n)$ is a coNP-complete language, while $\cup_{n\geq 2} \MQ_1(n)$ lies in DP. It is known that $\lim_{n\to \infty} |\MQ_1(n)|/|\MQ_0(n)|=1$. Here we prove the explicit finite-$n$ bounds \[ |\MQ_0(n)|<|\MQ_1(n)| \le \left(1+\frac{1}{2^n-1}\right)|\MQ_0(n)|, \] More generally, let $Q_d$ be the space of polynomial functions $(\FF_2)^n\to\mathbb F_2$ of degree at most $d$, and let $\alpha_k$ count square systems in $(Q_d)^n$ having exactly $k$ solutions. Then \[ \alpha_0<\alpha_1 \le \left(1+\frac{1}{2^n-1}\right)\alpha_0\,, \qquad 2\le d\le n \,. \] The proof combines matroid and coding-theoretic methods. We interpret $(\FF_2)^n$ as the ground set of the evaluation matroid of $Q_d$, express $\alpha_0$ and $\alpha_1$ through characteristic polynomials, and use a Whitney-type sign-reversing involution to show that the only terms that can push $\alpha_1-\alpha_0$ below $\alpha_1/2^n$ come from the elements of a matroid port. These are identified with minimal-support words of the Reed–Muller code $\RM(n-d-1,n)=\RM(d,n)^\perp$; the required estimate then follows from the MacWilliams identity, the minimum-distance bound $2^{d+1}$, and the even-weight structure of the code.

Gheibi, Jorgensen and Takahashi recently introduced the quasi-projective dimension, a homological invariant that extends the classical projective dimension. In this paper, we investigate this invariant from the perspective of Gorenstein homological algebra. First, we show that the quasi-projective dimension coincides with the Gorenstein projective dimension under certain conditions. Second, we introduce and study the quasi-Gorenstein projective dimension as a Gorenstein analogue of the quasi-projective dimension.

We construct a symmetric numerical semigroup $\Gamma$ of multiplicity $56$, embedding dimension $26$, and Frobenius number $181$ such that, over every field $k$, the one-dimensional Gorenstein local domain $R=k[t^\Gamma]_{\mathfrak{m}}$ has a nonprincipal ideal $I=(t^{56},t^{70})R$ for which $I\otimes_R\operatorname{Hom}_R(I,R)$ is torsion-free. This gives a counterexample to the Huneke-Wiegand conjecture in its general Gorenstein formulation. The proof reduces to an explicit finite sumset identity, interpreted through the Huneke-Iyengar-Wiegand colon criterion and Leamer's tensor-torsion formula. We also identify the tensor product with the trace ideal of $I$, compute $\operatorname{End}_R(I)=R[t^{101},t^{107}]$ and its conductor, and show that the counterexample persists under completion. A separate graded tensor-graph calculation and exact, standard-library Python verifiers accompany the paper. We explain how the example differs from the field-extension construction of Christensen, Gerko, and Iyengar.

In this article, we study the relations between algebraic models for the MinRank problem, which is a well-known NP-hard problem. We establish connections between the minors modeling, the Kipnis-Shamir modeling, and the support minors modeling, providing insights into their equivalences and differences. Our results contribute to a deeper understanding of the algebraic structures underlying the MinRank problem and may inform future research on efficient algorithms for solving it.

Let \(G\) be a finite simple graph on \(n\) vertices, and let \(I_c(G)\) be its complementary edge ideal. We determine \(\reg(I_c(G)^{(t)})\) for every \(t\geq1\) in terms of the number \(c(G)\) of nontrivial connected components of \(G\) and the presence of a \(K_2\)-component. Consequently, \(\reg(I_c(G)^{(t)})\leq \reg(I_c(G)^t) \) for all \(t\geq1\), with equality for every \(t\) if and only if either \(c(G)=1\), or \(c(G)=2\) and \(G\) has a \(K_2\)-component. We also characterize Serre's condition \((S_2)\) for \(R/I_c(G)\), and classify the graphs for which \(R/I_c(G)^{(t)}\) is Cohen-Macaulay for every \(t\geq1\).

We prove a Frobenius-dilation inequality for Hilbert functions of ideals generated by powers of fixed homogeneous forms in characteristic zero. Via Emsalem–Iarrobino inverse system duality, it yields quantitative fat-point postulation estimates and a new proof of the recent theorem of Ha–Sivakumar resolving Demailly's conjecture for arbitrary finite point sets. A fixed-characteristic version controls tangent spaces to varieties of powers and, together with plane interpolation and secant-theoretic results, proves complete nondefectivity of $V^k_{2,d}$ for all $k\ge3$ and $d\ge2$; in particular, it settles the ternary Waring problem for generic $k$-rank conjectured by Lundqvist–Oneto–Reznick–Shapiro (suggested by Ottaviani). We also show that, for a prime power exponent $q$, any failure of Nicklasson's conjecture in three variables is confined to $2q-2$ consecutive degrees.

We classify all filtered deformations of polynomial rings in three variables. We use our characterization to answer a special case of a question of Etingof by showing that if $F$ is an algebraically closed field of positive characteristic and $A$ is an $F$-algebra with a filtration by finite dimensional subspaces whose associated graded ring is a polynomial ring in three variables then $A$ satisfies a polynomial identity.