Papers

By a result of Backelin, Veronese subrings of a standard $\mathbb{Z}$-graded algebra are eventually Koszul. Davis, Erman, and Martinova recently conjectured that the analogous statement holds for Veronese subrings of polynomial rings with positive nonstandard $\mathbb{Z}$-gradings. We disprove this conjecture. For the weighted polynomial ring $\mathbb{K}[x_1,x_2,x_3,x_4]$ with weights $(1,4,7,9)$, we prove that the $(9k+12)$-th Veronese subring is not nonstandard Koszul for every $k\geq1$. Our counterexamples arise from certain arrangements of lattice points, which we call cubic obstruction configurations. Each such configuration produces a minimal cubic generator in the defining ideal of the corresponding associated graded ring. This construction yields counterexamples in $n$ variables for all $n\geq4$. Moreover, we prove that the set of primitive four-variable weight vectors for which eventual nonstandard Koszulness fails has positive density. For a fixed three-variable grading, our cubic obstruction can occur at only finitely many Veronese indices, leaving that case open.

Let $k$ be an algebraically closed field of characteristic zero. Degree-four rational maps on $\mathbb{P}^1$, up to conjugation, correspond to pairs of binary forms $(F,G)\in V_5\oplus V_3$. The associated invariant ring is the joint invariant ring $\mathcal{R}_{5,3}=k[V_5\oplus V_3]^{\mathrm{SL}_2}$. We determine a minimal generating set for $\mathcal{R}_{5,3}$, consisting of fifty explicit joint transvectants of degrees at most $18$. As consequences we describe the null cone of $V_5\oplus V_3$, construct six explicit absolute invariants that generate the function field of the moduli space $\mathcal{M}_4^1$, and give an effective criterion for determining conjugacy of degree-four rational maps.

We determine the Hilbert series of ideals generated by $d$'th powers of $n+2$ general linear forms in $n$ variables, to give upper bounds on the degree of the Hilbert series of ideals generated by $d$'th powers of $n+k$ general linear forms for $k>2$. This allows us to show that the Iarrobino-Fröberg Conjecture fails for all $n$ large enough. We also determine the degree of the Hilbert series for the ideal generated by $d$'th powers of $n+3$ general linear forms, for some values of $n$, and give counterexamples to a conjecture on the failure of the Weak Lefschetz Property for ideals generated by sufficiently large powers of general linear forms. Moreover, we determine the Hilbert series of the ideal generated by two generic quadratic forms in the exterior algebra on an even number of generators.

Based on work with Greenfeld and with Ziegler, Tao showed a concatenation result that if a map $f : \mathbb{F}^2 \to \mathbb{F}$ is affine-linear on every line parallel to the coordinate axes, and on all lines with a fixed nonzero slope (where the field $\mathbb{F}$ has size $> 2$), then $f$ is affine-linear on $\mathbb{F}^2$. We extend this from $\mathbb{F}^2$ to $\mathbb{F}^n$ and obtain a tight minimum number of additional lines needed – $N = \binom{n}{\lfloor n/2 \rfloor}$ – for every field $\mathbb{F}$ with $3 \leqslant n < |\mathbb{F}|$. The proof is constructive and shows a stronger result: the existence of a universal family of $0$-$1$ matrices of size $\binom{n}{k} \times \binom{n}{k}$ (one for each pair $0 \leqslant k \leqslant n$), which are indexed by ballot sets and are unimodular over all unital commutative rings. We also show a second tightness: of the assumption $n < |\mathbb{F}|$. Else, there exist multi-affine maps $f$ which are affine-linear on every line through the origin, but not affine-linear globally on $\mathbb{F}^n$. More strongly, we prove this dichotomy – including the bound of $N$ – over all integral domains, or Noetherian (e.g.\ finite or Artinian) rings, or products of these. This yields a novel numerical invariant for affine-linearity, for every product of Noetherian rings and integral domains.

Problem F of \cite{FLOS18} asks whether an ideal generated by $r$ generic $\mu$-power forms, where $\mu\vdash d$ is a non-pure partition, has the same Hilbert series $G_{n,d,r}=\big[(1-t^d)^r(1-t)^{-n}\big]_+$ as an ideal generated by $r$ generic forms of degree $d$. We show that the answer is negative, and we determine exactly when the natural obstruction is available. Our main tool is an apolarity inclusion that converts a distinguished large part of $\mu$ into a fat-point linear system in the dual projective space. This yields a single master obstruction (Theorem \ref{thm:master}): a counterexample exists whenever, for $r$ general points of $\PP^{n-1}$, the homogeneous system $\LL(q;m^r)$ is non-empty while $\LL(q;(m-a)^r)$ has non-positive virtual dimension. We complement this with a positive result (Theorem \ref{thm:small-r}): by Stanley's theorem, Problem F has a positive answer for every $\mu$ as soon as $r\le n+1$. For ternary forms, the obstruction produces counterexamples precisely for $5\le r\le 8$, with extremal curves given by the conic through five points and products of $(-1)$-curves on del Pezzo surfaces. It is vacuous unconditionally for $r\le 4$ and $r=9$, and, assuming Nagata's conjecture, for every non-pure $\mu$ when $r\ge 10$. We also obtain counterexamples in four variables for $r=9$. An exact finite-field certificate shows that $d=14$, $\mu=(13,1)$ is the first failure for five ternary generators.

We characterize all first cosyzygy modules of a given finitely generated module over a local ring. We do this by introducing the notion of a canonical cosyzygy module, and showing how other cosyzygy modules can be obtained from this module. As an application, we characterize Gorenstein rings via properties of canonical first cosyzygy modules.

We study the Marshall quotient \[ M(n)=M(\mathbb Z/n\mathbb Z) \] obtained from the ring of integers modulo $n$ by quotienting by the square classes of non-zero-divisors. Using elementary arithmetic of square classes modulo prime powers and the Chinese Remainder Theorem, we give an explicit description of these quotients and classify several of their structural properties. We determine when the quotient relation is arithmetically elementary, when $M(n)$ is hyperbolic, when it can be real reduced, and when it can be formally real. We also analyze the subset of invertible classes together with zero, proving exactly when it is a submultiring, when it is a hyperfield, and when it is hyperbolic. The results provide a finite family of test examples for questions connecting multirings, special hyperfields, real semigroups, and abstract quadratic-form theory.

We establish formulae for point counts of cluster varieties of cluster algebras of marked surfaces, possibly with punctures. We then establish formulae for the number of non-deep points in these cluster varieties over $\mathbb{F}_2$, which gives us the number of algebraic tori necessary to cover the cluster manifold over $\mathbb{F}_2$. We also show that these formulae satisfy certain recurrence relations.

It follows by Puuska that, over a commutative Noetherian ring $R$, a morphism $i : M \rightarrow I$ of $R$-modules is an injective envelope if and only if its Matlis dual $Hom_R(i, E)$ is a flat cover for some injective cogenerator $E$, and equivalently for every injective $R$-module $E$. In this paper, we will generalize this result to non-commutative Noetherian algebras.

We discuss an example of a rigid non-principal ideal in a one dimensional (commutative) Gorenstein domain, which contradicts a conjecture of C. Huneke and R. Wiegand. The construction and its analysis were discovered by Codex, when prompted by one of the authors to verify the conjecture or find a counterexample. A proof of the conjecture, also discovered by Codex, is presented when the ring is equicharacteristic and its embedding dimension is at most three.

We give a complete characterization of the positive integer pairs $(e,m)$ that occur as the (embedding dimension, multiplicity) of a complete intersection numerical semigroup, thereby settling an open problem in the theory of relations of numerical semigroups. We prove that there exists a complete intersection numerical semigroup $\Gamma$ with $e(\Gamma)=e$ and $m(\Gamma)=m$ if and only if $(e,m)=(1,1)$ or $e\ge 2$ and $m\ge 2^{e-1}$.

Motivated by refinements of Kunz's theorem involving the growth of Betti numbers, we study Betti and Bass numbers associated to the relative Frobenius. Under appropriate hypotheses on a local ring homomorphism $\varphi \colon R \to S$, we obtain bounds on the growth of the Betti and Bass numbers of relative Frobenius pushforwards of homologically finite complexes in terms of the corresponding invariants of the residue field of the closed fiber $S/\mathfrak m S$. These results extend previous work on Betti numbers to coefficients and provide a dual perspective via Bass numbers, contributing to the study of how Frobenius actions encode information about ring homomorphisms.

Let $\mathcal E=(L/K,v)$ be a finite Galois extension of henselian valued fields. We study the ramification ideals $I_H$, for subgroups $H\leq {\rm Gal}(L/K)$, when the Galois group is a product of subgroups $H_i$ such that $L/K_{H_i}$ is pure (depth one). We first recall an explicit formula for ramification ideals of pure extensions and use it to obtain a lower bound for the ideals attached to arbitrary subgroups of a product of pure subgroups. A natural question is whether every $I_H$ coincides with one of the ideals coming from the pure factors. We show that this is not true, already for a defectless extension with Galois group $C_p\times C_p$. The counterexample is a compositum of two Artin–Schreier extensions with different ramification breaks; the failure comes from choosing a decomposition which is not compatible with the ramification filtration. Motivated by this example, we introduce ramification-adapted decompositions and prove a filtration-theoretic substitute for the conjectural statement in elementary abelian $p$-extensions. Since every flag of $\mathbb F_p$-vector spaces admits an adapted basis, every elementary abelian $p$-extension admits such a decomposition, and every subgroup ideal is represented by one adapted cyclic factor. If the adapted factors are pure, this representation can be written in the distance-set form occurring in the original conjecture. We also prove that, for an adapted decomposition $\mathcal G=H_1\times\cdots\times H_r$, all ramification ideals are principal if and only if every degree-$p$ extension $L/K_{H_i}$ is defectless. Finally, we discuss the degree-$p^2$ example constructed by Kuhlmann in \cite[Section 3.5]{Topics}. Consequently every basis is ramification-adapted, while principality of all ramification ideals still does not characterize defectlessness.

Let $A$ be a commutative ring and let $I$ be an ideal. An element $a\in A$ is a regular denominator for $I$ when multiplication by $a$ on $A/I$ is injective; we denote the set of all such elements by $S_I$. We study how the common regular denominators for two ideals $I$ and $J$ are related to $I\cap J$ and $I+J$. This yields a particularly simple description when $I$ and $J$ are comaximal. Over Noetherian rings, the same viewpoint classifies denominator-equivalence classes by finite nonempty antichains of prime ideals, gives bounds for the associated primes of an intersection and leads to a local length identity. Furthermore, we show that flat base change preserves regular denominators, faithful flatness reflects them, and finite locally free quotients have fiberwise regular loci defined by determinants satisfying a multiplicative Mayer-Vietoris formula. Our results provide alternatives to primary-decomposition computations in many concrete situations.

Understanding the features captured by the hidden layers of neural networks is a fundamental challenge in machine learning, despite their widespread success across various classification problems. In this work, we propose an algebraic framework for examining neural networks that model classification problems. Certain results, such as the correspondence between the neural network and neural ideals, algorithms for computing the neural ideals, and a stabilization theorem that enables approximation of the neural ideals, are first established. As an application to the framework, we present algorithms to identify and interpret the features captured by each hidden-layer neuron. Along with these theoretical developments, the practical performance has been demonstrated on the MNIST digit dataset, and the results highlight the pivotal role of neural ideals as a mathematical and computational tool for analyzing the features captured by neural networks. Further, we develop an interactive software that builds on the presented framework to visualize the features captured by each neuron. This tool is available at https://github.com/yvs1967/neural-network-representation-explorer

We consider systems of homogeneous multivariate polynomials equivariant under the complex reflection group $G(r,n) = (\mathbb{Z}/r\mathbb{Z})^n \rtimes S_n$. Using divided differences indexed by partitions of $n$, we establish a decomposition formula expressing the resultant of such a system as a product of resultants of smaller, partition-indexed subsystems. Combining this with the classical resultant–discriminant relation, we show that the discriminant of a $G(r,n)$-invariant homogeneous polynomial splits explicitly into a product of resultants of smaller subsystems, considerably easier to compute; we illustrate both decompositions with worked examples.

Let $G$ be a finite simple co-chordal graph with at least one edge, and let $I(G)$ be its edge ideal. Over an arbitrary field, we prove that the symbolic powers of $I(G)$ satisfy $\operatorname{reg} I(G)^{(s)}=2s$ for every $s\ge1$. Thus every symbolic power of $I(G)$ has a degree resolution. Using Takayama's formula and clique trees, we reduce the regularity problem to a topological one. Finite convex geometry then shows that nontrivial homology implies a suitable set of leaves, and a weighted counting argument gives the required regularity formula. Finally, we show that symbolic powers of co-chordal edge ideals are not necessarily componentwise linear.

In this paper, we study the derivations of an irreducible plane algebroid curve $R:=\frac{k[[X,Y]]}{(f)}$, which is not regular. Since the normalization of $R$ is isomorphic to $k[[t]]$, every $k$-derivation of $R$ is induced from a $k$-derivation of $k[[t]]$, which are of the form $a(t)\frac{d}{dt}$ for some $a(t)\in k[[t]]$. We establish a lower bound on the $t$-order of a nonzero power series $a(t)$ for $a(t)\frac{d}{dt}$ to induce a derivation of $R$ in terms of the multiplicity of $R$. We also prove a related result for the value semi-group of such curves.

Let $R$ be a commutative Noetherian ring, let $\mathbf{x}=x_1,\ldots,x_n$ be an $R$-regular sequence, and let $\mathbf{y}=y_1,\ldots,y_m$ be a sequence of elements of $R$. Put $I=(\mathbf y)$. Let $\mathcal S$ be a Serre subcategory of the category of $R$-modules. We consider the double complex obtained from the Koszul co-complex with respect to $\mathbf x$ and the Čech complex with respect to $\mathbf y$. Using the two spectral sequences associated with this double complex, we prove that \[ Ext_R^i(R/(\mathbf x),H_I^j(R))\in\mathcal S \quad\text{for all }i,j\in \mathbb N_0 \] implies \[ H_I^j(R/(\mathbf x))\in\mathcal S \quad\text{for all }j\in\mathbb N_0. \]

We provide a positive answer to a conjecture proposed by Tohǎneanu and Van Tuyl regarding the minimum distance of codes whose underlying set of points is a reduced complete intersection. Despite the technical nature of the conjecture, we show that it follows directly from a not-well-known refinement of the classical Bézout bound for overdetermined polynomial systems. For completeness, this paper presents a self-contained proof of this refined bound. Furthermore, we show that using the same approach, it is possible to obtain a bound on the generalized Hamming weights of such a code and, more generally, to control the minimum distance of the codes obtained by evaluating forms of degree $d$ on the points of a zero-dimensional complete intersection.

Computing eigenvalues and eigenvectors is at the heart of the solution of many non-linear problems. For instance, finding the roots of polynomial systems reduces to computing joint eigenvectors of operators of multiplication. Similarly, tensor decomposition can be performed via the joint diagonalization of submatrices of the Catalecticant of the tensor. We describe and illustrate symbolic-numeric methods for computing the solutions of these algebraic problems from the computation of joint eigenvectors of commuting operators, and for analysing their multiplicity structure, as well as their implementation in the package AlgebraicSolvers.jl.

For a $C_4$-free graph $G$ on $n$ vertices — one with no induced cycle on four vertices — we study the two-sided extremal problem for the triangle density $\tau$: How large and how small can $\tau$ be for given edge density $\varepsilon$ and clique-number density $\kappa = \omega(G)/n$? We give lower and upper bounds for $\tau$ in terms of $\kappa$ and $\varepsilon$. The two bounds sandwich $\tau$, and their compatibility forces a lower bound for $\kappa$ in terms of $\varepsilon$. When the clique complex of $G$ is $2$-Leray over a field $\Bbbk$, the resulting bound on the clique-number density lies between the previous best $C_4$-free bound and the sharp chordal bound. It improves on the former for every $\varepsilon \in (0,1)$. The lower bound is elementary. The upper bound is homological, obtained by passing to the Stanley–Reisner ring of the clique complex. When the complex is $2$-Leray, its Betti table has at most two linear strands. The two first entries in the first strand encode edge and triangle densities, and the strong structural form of a Boij–Söderberg decomposition constrains what these entries can be, yielding the upper bound. For $2$-Leray graphs with no holes in the range $[4,g]$ we give a conjecturally sharp bound. We further ask questions concerning the triangle bound for any $C_4$-free graph.

The classical Veronese formula of Goto–Watanabe identifies the canonical module of a Veronese subalgebra with the corresponding Veronese module. We prove the analogous compatibility for graded duals under a module-finite extension whose upper ring is equidimensional, satisfies Serre's condition $(S_2)$, and has dimension at least two, and obtain the formula for the $b$-invariant. This yields explicit two-sided bounds for canonical traces: the lower bound is controlled by the Loewy length of the canonical-trace quotient, whereas the upper bound is governed by the $a$- and $b$-invariants, with an $S_2$-ification refinement for unmixed rings. We also characterize when all sufficiently large Veronese subalgebras of standard graded level algebras are nearly Gorenstein when the $a$-invariant is negative. Applications include Segre products, Stanley–Reisner rings and determinantal rings.

We show that tree tensor network varieties, including tensor train varieties, are general Markov models associated to spaced trees. This allows us to prove that the prime ideals of these varieties are generated by minors of matrix flattenings. In the case of tensor train varieties, we discuss whether these minors form a Gröbner basis and provide a combinatorial method to compute the degree for order $3$ tensor trains.

Let $\mathsf{CI}_{\leq c}$ denote the property of being a complete intersection of codimension at most $c$. Although the regular, complete intersection, Gorenstein, and Cohen–Macaulay properties satisfy the Nagata criterion (NC), we prove that $\mathsf{CI}_{\leq c}$ does not satisfy (NC) for any $c \geq 1$. We first construct a counterexample for hypersurfaces and then obtain counterexamples for arbitrary $c$ using square-zero extensions. We also introduce three conditions for a property of Noetherian local rings and show that, under stability with respect to localization and reduction by suitable regular sequences, (NC) is characterized by a lifting property across normally flat nilpotent thickenings. Nevertheless, we recover the expected openness result: the $\mathsf{CI}_{\leq c}$-locus is open for every Noetherian ring satisfying $\mathsf{Reg}$-Q0, and hence for every quasi-excellent ring. Finally, for every quasi-compact excellent scheme $X$, we prove that the subset $\left\{x \in X \mid \operatorname{edim}(\mathcal{O}_{X,x}) - \dim(\mathcal{O}_{X,x}) \leq n \right\}$ is constructible for every $n \in \mathbb{N}$, although the function $x \mapsto \operatorname{edim}(\mathcal{O}_{X,x}) - \dim(\mathcal{O}_{X,x})$ is not upper semicontinuous in general.

Let $W$ be a henselian valuation ring with fraction field $L$, residue field $k$, and value group $\Gamma_W$. Let $N=\ell^\nu$ be invertible in $W$. Choose an ordered $\mathbf Z/N$-basis $B$ of $\Gamma_W/N\Gamma_W$. Products of suitable classes define an equivalence of complete filtered spectra $$ \bigoplus_{\substack{J\subseteq B\\J\text{ finite}}} \Sigma^{|J|}\operatorname{Fil}_{\mathrm{mot}}^{\bullet-|J|} K(k;\mathbf Z/N) \xrightarrow{\simeq} \operatorname{Fil}_{\mathrm{mot}}^{\bullet} K(L;\mathbf Z/N) $$ The summand indexed by $\varnothing$ is the generic restriction map, after the rigidity equivalence $K(W;\mathbf Z/N)\simeq K(k;\mathbf Z/N)$, and is therefore split injective. Independently of this splitting, excision yields a lifting theorem for regular henselian pairs. In particular, this yields finite-coefficient Gersten injectivity for noetherian henselian regular local rings. Applications to completions along regular primes give relative and sometimes nonhenselian examples. More generally, if $P$ is a Prüfer ring and $R$ is a henselian local ind-smooth $P$-algebra, then $R$ is a domain and $K_n(R;\mathbf Z/N)\to K_n(\operatorname{Frac}(R);\mathbf Z/N)$ is injective for every $n$, provided $N\in R^\times$. These injectivity consequences extend from prime-power to arbitrary finite invertible coefficients by primary decomposition.

Let $M$ be a simple matroid and let $B(M)$ be the graded Möbius algebra of its lattice of flats. The ordered-basis weights of flats define an inner product for which the adjoints of atom multiplication become ordinary coordinate derivatives under the basis-polynomial realization. From this, we construct a canonical global lowering operator $D_\beta$ which acts as ordinary differentiation on a canonical “radial” copy of a truncated polynomial algebra. Allowing both $D_\beta$ and the coordinate derivatives to act produces a graded cyclic module with Hilbert series \[ H_{\beta,M}(q)=\sum_{k=0}^r h_k^\beta(M)q^k. \] We give examples of matroids with the same Derksen $\mathcal G$-invariant and the same classical apolar Hilbert series but different $H_\beta$. Hence $H_\beta$ cannot be the restriction to simple matroids of a valuative matroid invariant. We conjecture that $H_\beta$ is log-concave and top-heavy in differential degree. For the generalized theta family containing Larson's counterexample to Whitney log-concavity, we compute the first four coefficients and prove the critical log-concavity inequality. Exact computation verifies both conjectures for all $950$ simple matroids on eight elements.

The main purpose of this paper is to provide a simple proof of the isometry theorem for one-parameter persistence modules, in which a matrix representing one morphism of an interleaving is reduced and the pivots determine a matching between barcodes. This approach applies to persistence modules of finite type indexed by the reals, and the more general statement for q-tame modules can then be deduced from it using approximations of q-tame modules. The similarity between this use of matrix reduction and that in the persistent homology algorithm motivates some further development of matrix computations for persistence modules, formalized by a category of barcodes in which the morphisms are equivalence classes of matrices. A method for computing induced maps on persistent homology is provided using matrix operations that fit naturally into persistent homology computations, making functorial persistent homology barcodes computable. A method is also given for computing persistent homology barcodes and morphisms when chains do not necessarily have infinite death times.

Let $H$ be a unimodular hypergraph with cover ideal $J(H)$. We prove that the local $v$-numbers of $J(H)^t$ are linear in $t$ for all $t\ge1$. We further show that the global $v$-number of $J(H)^t$ is linear in $t$ for all $t\ge n-1$. Finally, we prove that the global $v$-number of the powers of the cover ideal of any tree is linear in $t$ for all $t\ge1$.

We prove that if two parameterizations of a complete reduced noetherian curve over an algebraically closed field agree modulo a sufficiently large power of the maximal ideal, then the two parameterizations are equivalent. This strengthens some bounds from Greuel and Pfister. In addition, we prove that if two reduced and irreducible curve singularities are isomorphic modulo sufficiently high (and identical) powers of their respective maximal ideals, then the completions of the two curves are isomorphic, and the isomorphism of the full completions agrees with the original isomorphism modulo some lower power of the maximal ideals. We provide a new and better bound on the lower power, strengthening the bound in Hironaka