Papers

We give an example of a finite ring for which the set of polynomials inducing the zero function fails to be a two-sided ideal, disproving a conjecture of Werner.

The determinant of a finite semigroup is closely related to the Frobenius property of its corresponding semigroup algebra, and it plays a significant role in coding theory applications. In this paper, we study determinants of twisted contracted semigroup algebras over finite fields. We extend Steinberg's determinant theory to this setting by establishing a Frobenius criterion for twisted contracted semigroup algebras over finite fields. We also develop a structural theory for two-generated twisted contracted monoid algebras. In particular, we identify classes for which the determinant of the associated matrix depends only on its zero–nonzero pattern, and is therefore independent of the values of the cocycle. Consequently, it suffices to verify the determinant condition for the trivial character, from which the corresponding result for all characters follows. Under suitable assumptions, these semigroups are completely determined by a small set of defining identities. Motivated by the character-theoretic construction of homogeneous weights on finite Frobenius rings, we construct homogeneous weights for Frobenius twisted contracted semigroup algebras and investigate the non-Frobenius case. In particular, we characterize the principal ideals on which the character-average weight satisfies the averaging property of homogeneous weights and thereby extend the classical notion of homogeneous weight to a prescribed family of principal ideals.

In this article, we characterize the isotropy groups of certain special monomial and Jouanolou-type derivations of polynomial rings over fields of characteristic zero. Under suitable conditions, we determine the structure of these isotropy groups.

We give a convex-geometric formula for the multiplicity sequence of a monomial ideal in terms of mixed volumes of polytopes constructed from its Newton polyhedron. We also construct a counterexample to a conjecture of Achilles and Manaresi proposing a different volume formula for the multiplicity sequence. Finally, we derive a mixed-volume formula for the mixed multiplicities of arbitrary monomial ideals.

Let $R$ be a commutative Noetherian ring. We prove that if the one-variable formal power-series ring $R[[x]]$ is a unique factorization domain, then so is the two-variable formal power-series ring $R[[x,y]]$. This resolves a question raised by Bayart in 1973 for Noetherian coefficient rings. The proof uses the divisor theory of Noetherian normal domains, expressed through finite rank-one reflexive modules.

In this paper, we prove that the Artinian Gorenstein $\mathbb{K}$-algebra $A_{F_s}$ of codimension $n$, socle degree $d$ and Macaulay dual generator $F_s := \ell_1^d + \dots + \ell_s^d \in \mathbb{K}[X_1, \dots, X_n]$ where $\ell _1, \cdots , \ell_s$ are general linear forms satisfies the strong Lefschetz property (SLP). This result allows us to study whether the Waring rank of $F_s$ is exactly $s$. Furthermore, we show that $A_{F_s}$ is the doubling of a suitable 0-dimensional scheme $Z_{F_s}$ in $\mathbb{P}^{n-1}$, the so-called tight annihilating scheme of $A_{F_s}$, and we compute the minimal free resolution of $A_{F_s}$ in terms of the minimal free $R$-resolution of $I(Z_{F_s})$. Finally, we determine the linear general Jordan type of $A_{F_s}$.

Let $A$ be a finitely generated commutative algebra over a field $\mathbb K$ of characteristic zero. We prove that every finitely generated Lie subalgebra of $\operatorname{Der}_{\mathbb K}(A)$ whose elements are locally finite on $A$ is finite-dimensional. Consequently, for a Lie subalgebra generated by finitely many locally finite derivations, the following are equivalent: it is finite-dimensional, it acts locally finitely on $A$, and all its elements are locally finite. A key ingredient is a second theorem: every Lie subalgebra of $\operatorname{Der}_{\mathbb K}(A)$ whose elements are locally nilpotent is solvable; when $A$ is reduced, its derived length is at most $\dim A$, and this bound is sharp. In general, the $(\dim A)$-th term of its derived series is nilpotent, of class bounded in terms of $A$ only. We also prove that every solvable Lie subalgebra of $\operatorname{Der}_{\mathbb K}(A)$ generated by finitely many locally finite derivations is finite-dimensional and acts locally finitely on $A$. For an affine variety $X$ over an algebraically closed field, we deduce that a subgroup of $\operatorname{Aut}(X)$ generated by finitely many connected algebraic subgroups is algebraic if and only if every element of the Lie algebra generated by their tangent algebras is locally finite. For two unipotent one-parameter subgroups, this gives an answer to a 2005 problem of Popov. Our results also characterize polynomial control systems admitting an exact finite-dimensional bilinear realization by polynomial observables containing the state coordinates. The proof uses derivations vanishing to second order along a suitable finite subscheme, on which local finiteness forces local nilpotence.

Let $(p,q)$ be a Jacobian pair. We show that the real Jacobian conjecture holds if the degree of $p$ is 7 and the highest degree homogenous part is of the form $\alpha x^7 + \beta x^6y$ for $\alpha^2+\beta^2 \neq 0$. We then show that there are no atypical Jacobian pairs such that $\mathrm{deg} \, p =7$ and $\mathrm{deg}\, q$ is even and coprime with 7.

Let $R$ be a $d$-dimensional Noetherian local ring with maximal ideal $m_R$ and let $I$ be an ideal of $R$. The epsilon multiplicity $\epsilon(I)$ of $I$ is defined by Ulrich and Validashti in \cite{UV} as a limsup of normalized lengths. We show in this paper that the epsilon multiplicity is always a limit of these normalized lengths. Specifically, $$ \epsilon(I)= \lim_{n\rightarrow\infty} \frac{\ell_R((I^n)^{\rm sat}/I^n)}{n^d/d!}. $$ We prove that the epsilon multiplicity satisfies an associativity formula, analogous to that of the ordinary multiplicity. We prove these results by comparing the epsilon multiplicity to a suitable limsup of normalized Amao multiplicities, the Amao epsilon multiplicity $a\epsilon(I)$. We show that the epsilon multiplicity is always equal to the Amao epsilon multiplicity, which is itself a limit.

Persistent tensors form a recursively defined class adapted to the substitution method and provide nontrivial lower bounds on tensor rank. We study the behavior of symmetric persistent tensors under Kronecker products. We establish a global differentiation identity expressing the Hessian matrix of a Kronecker product of homogeneous polynomials in terms of the Hessian matrices of its factors; although no corresponding determinant identity holds globally, the Hessian polynomials factor exactly at decomposable points. This yields a multiplicative formula for the distinguished Hessian coefficients of isobaric forms and closure under Kronecker products for symmetric persistent tensors that are isobaric of the distinguished weight, including iterated products and powers. Combined with the classification in small dimensions, this implies closure when both factors have dimension at most three, and for persistent cubics when both have dimension at most four. We further give sufficient closure criteria via simultaneous strict triangularizability of normalized Hessian spaces, for cubics and then arbitrary degree, and show that the distinguished isobaric class satisfies this condition. Finally, we show that symmetric persistence is not preserved in general by constructing a persistent cubic $f\in\operatorname{Sym}^3\mathbb{C}^{12}$ such that $f\boxtimes f\in\operatorname{Sym}^3\mathbb{C}^{144}$ is not persistent. The obstruction occurs at a nondecomposable point, showing that the Hessian factorization on the Segre variety does not extend to the full tensor product space.

We show that any $d$-dimensional local ring $A$ with a dualizing complex, $\mathrm{depth} A=d-1$, and cyclic deficiency module $K^{d-1}(A)$ admits a maximal Cohen–Macaulay module. It is constructed as the unique nonzero cohomology module of the cone of the derived morphism induced by a surjection $A\to K^{d-1}(A)$. When $A$ is quasi-Gorenstein, this module is identified with the first syzygy of the canonical module $\omega_{A/xA}$, for any $x\in\operatorname{ann}_A K^{d-1}(A)$ that is regular on $A$. This recovers a theorem of Tavanfar and Shimomoto in the $3$-dimensional quasi-Gorenstein case with $K^2(A)\cong k$. We also give examples of section rings satisfying the hypotheses of our theorem.

We introduce mutation-preserving generalized cluster algebras, for which the generalized cluster mutation in each direction is independent of the seed in the mutation equivalence class. We classify all irreducible generalized cluster algebras with this property. Then, a Markov-type Diophantine equation $x^2+y^2+z^2+2yz=kxyz$ is studied, which has a structure of the mutation-preserving generalized cluster algebra. We prove that positive integer solutions exist if and only if $1\leq k\leq 5$ and determine all their orbits under the associated generalized cluster mutation groups. In particular, multiple orbits occur for each $k=1,3$, whereas the solutions form a single orbit for each $ k=2,4,5$. We then classify all generalized Markov Laurent mutation invariants. As an application, a conjecture proposed by Chen-Li is proved, showing that every Laurent mutation invariant of irreducible sign-equivalent cluster algebras is essentially a polynomial in the corresponding basic invariant.

We prove that all (possibly infinitely generated) semi-Gorenstein-projective modules over a virtually Gorenstein Artin algebra are Gorenstein projective. We also establish a new criterion for determining the projective (injective) dimensions of modules over virtually Gorenstein Artin algebras. This enables us to show the validities of the Auslander-Gorenstein Conjecture, the (Strong) Nakayama Conjecture and Tachikawa's First Conjecture for virtually Gorenstein Artin algebras. Finally, we prove that the Auslander-Reiten Conjecture holds for virtually Gorenstein Artin algebras of finite CM-type.

A. A. Suslin and V. I. Kopeiko proved that the elementary linear, symplectic, and orthogonal groups are normal in the general linear group, symplectic group, and orthogonal group respectively. They also proved relative versions of these normality results with respect to an ideal of a ring. A. Bak, R. Basu, and R. A. Rao proved that the linear, symplectic, and orthogonal transvection groups are normal in the respective automorphism groups. These results are generalizations of the normality results by Suslin and Kopeiko in the setup of modules. In this paper, we prove stronger versions of the normality results proved by Bak, Basu, and Rao, which say that the relative transvection groups are normal in the respective automorphism groups.

Let $X$ be a smooth projective surface with $p_g=0$, and let $H$ be an ample divisor with $h^0(\mathcal{O}_X(H))\neq0$, $\chi(\mathcal{O}_X(H))\ge q$, and $h^1(\mathcal{O}_X(H))\neq0$. We prove that no rank two bundle $\mathcal{E}$ with $c_1(\mathcal{E})=3H+K_X$, with the Ulrich value of $c_2(\mathcal{E})$, and satisfying $h^0(\mathcal{E}(-H))=0$, can arise from an extension $0 \to \mathcal{O}_X(H+K_X) \to \mathcal{E} \to \mathcal{O}_X(2H)\otimes\mathcal{I}_Z \to 0$. Thus, for this natural Cayley-Bacharach construction, non-speciality of the polarization is necessary rather than merely convenient. We then study primary Burniat surfaces. We show that every ample and base point free divisor is non-special; consequently, every polarization carries a stable special Ulrich bundle of rank two, and the surface is strictly Ulrich wild with respect to every polarization. We also locate the special ample classes on three numerical rays through $K_X$, compute explicit families on these rays, and analyze a twisted-kernel variant of the construction. The degree bound underlying the non-speciality result overlaps with recent work of Y. Cho, while the global consequences and the no-go theorem are independent.

In this paper, we show that the tensor rank of the $5\times 5$ permanent tensor is exactly $16$ over fields of characteristic zero, by proving the lower bound matching the known upper bound. The $5\times 5$ permanent tensor is a symmetric tensor corresponding to the monomial $x_1x_2x_3x_4x_5$, whose Waring rank is known to be $16$. Previously, it was known, by the higher-order Koszul flattening and Fischer's formula, that its tensor rank is either $15$ or $16$.

We give a negative answer to a question of Avramov, Buchweitz and Şega by constructing a commutative local finite-dimensional non-Gorenstein algebra $R$ with $\operatorname{Ext}_R^1(D(R),R)=0$; this question is related to the first Tachikawa conjecture. We also give a counterexample to a conjecture of Huneke, Şega and Vraciu on Tor vanishing over commutative finite-dimensional algebras. Finally, we construct a finite-dimensional commutative local self-injective algebra $R$ over $\mathbb{F}_2$ and an indecomposable non-projective $R$-module $M$ such that $\operatorname{Ext}_R^1(M,M)=\operatorname{Ext}_R^2(M,M)=0$, related to the second Tachikawa conjecture and answering a question of Dao.

Let $R$ be an excellent local domain. $R$ is said to be $NBIM$ if $Tor_{i}^{R}(R^{+}, k) = 0$ for some $i\geq d:=\dim(R)$. Bhatt, Iyengar, and Ma ask if equi-characteristic zero $NBIM$ rings are regular. If $R$ is of positive characteristic, Asgharzadeh and Mahdavi conjecture that $Ext^{i}_{R}(k,R^{\infty}) = 0$ for some $i>d$ implies that $R$ is regular. It is an open question whether $R^{+}$ and $R^{\infty}$ are $\mathfrak{m}$-adically idealwise separated in positive characteristic, a condition from the `local criterion of flatness'. These are analogues of Kunz's theorem and intimately related to the homological conjectures and singularities in algebraic geometry. We apply a result of Avramov, Hochster, Iyengar, and Yao on contracting endomorphisms to make progress on the first two. We observe that it implies toric $NBIM$ rings are regular and solves the conjecture for $F$-pure rings. These improvements are inaccessible by previous techniques and give new and simple proofs of earlier results. In mixed characteristic, we show several linked results for perfectoid-pure rings. We show the third statement when there is $R\rightarrow S$ finite and flat on the punctured spectrum and $S$ is regular, this uses Cohen-Macaulayness of $S^{+}$.

For any ideal $I$ (whose projective dimension need not be finite) in a local Noetherian ring $R$, we show that, if the conormal module $I/I^2$ has a free summand given by $F$, the natural map from $\bigwedge F$ to $\text{Tor}_{*}^{R}(R/I,R/I)$ splits as algebras. To achieve this, we use dg algebra techniques and remodel results of André and Iyengar, proving them in setting of semi-free extensions, replacing the need for semi-free $\Gamma$-extensions. In this new setting, we show that free summands of the conormal module correspond to central elements of the relative homotopy Lie algebra $\pi^*(\varphi)$ and, lastly, we provide an explicit splitting morphism in lower degrees.

We study the Gorenstein homological dimension $Ghd_k G$ of groups $G$ which are of type $FP_{\infty}$ over a commutative ring $k$. Our main result shows that, over an arbitrary ring $R$, every finitely presented Gorenstein flat $R$-module is projectively coresolved Gorenstein flat. Consequently, for a group $G$ of type $FP_\infty$ over $k$ with sfli$k<\infty$, the three natural dimensions for $G$, namely Gorenstein homological, Gorenstein cohomological and projectively coresolved Gorenstein flat, all coincide. Building on this collapse, we establish a Gorenstein homological analogue of Fel$'$dman's theorem, a formula for iterated $m$-fold self-extensions of a group $N$ of type $FP_\infty$ over $\mathbb Z$, and a field-detection theorem, showing that the Gorenstein homological dimension of a group of type $FP_\infty$ over a principal ideal domain is realized after passing to a suitable field.

For a commutative Bezout ring R, we give a criterion, in terms of colon ideals with principal radical, for the lattice $\mathring{\mathcal{K}}(\mathrm{Spec}(R))$ of compact open subsets of $\mathrm{Spec}(R)$ to be a Heyting algebra. Bezhanishvili and Tressl showed that $\mathring{\mathcal{K}}(\mathrm{Spec}(C(T)))$ is pseudocomplemented whenever T is a basically disconnected compact Hausdorff space, and asked whether $\mathrm{Spec}(C(\beta\mathbb{N}))$ is actually an Esakia space. We show it is not: applying our criterion to $C(\beta\mathbb{N}) \cong \ell^\infty(\mathbb{N},\mathbb{R})$ produces a diagonal counterexample, and the same obstruction rules out $\beta D$ for every infinite discrete D. A grid-existence theorem for $\sigma$-complete Boolean algebras lets us push the construction to every basically disconnected compact Hausdorff space, settling the Bezhanishvili-Tressl question completely: for no infinite compact Hausdorff space T is $\mathrm{Spec}(C(T))$ an Esakia space.

A noetherian ring is called G-regular when all finitely generated Gorenstein projective modules are projective. In this paper, we study rings satisfying the stronger condition that all Gorenstein projective modules are projective, which we call strongly G-regular. We show that the notion of strongly G-regular rings is closely related to that of quasi-dominant rings introduced by Takahashi and to the covariant/contravariant finiteness of a certain thick subcategory. We also answer a series of questions due to Chen in the negative, showing that the Gorenstein projective analogue of the Auslander-Ringel-Tachikawa theorem fails even for commutative local artin algebras which are weakly Gorenstein in the sense of Ringel and Zhang.

Let $(R,\mathfrak m)$ be an unramified regular local ring of mixed characteristic $(0,p)$ and dimension $d$ and let $I\subseteq R$ be an ideal. We prove that $depth(R/I)\geq 3$ implies $cd(I)\leq d-3$, and if $R$ is essentially of finite type over a DVR, then $depth(R/I)\geq 4$ implies $cd(I)\leq d-4$. More generally, $H_I^j(R)$ is a $\mathbb{Q}$-vector space whenever $j>d-depth(R/I)$, thus vanishing of local cohomology in this range is determined completely by the characteristic zero fiber.

We develop an algebraic framework for central configurations of the $n$-body problem with homogeneous potentials, grounded in the exterior algebra of the configuration space and the normalized shifted Brehm–Wintner–Conley (BWC) matrix $S$. Relating the kernel of $S$ to the Plücker coordinates of the configuration, we generalize the determinantal equations obtained by Williams (1938) for the planar five-body problem to central configurations of any dimension and any number of bodies, and derive the Dziobek–Williams equations $\det(S_I^J)=\kappa\, z_I z_J$, which exhibit the compound matrix $S^{(t)}$ as a rank-one matrix. Introducing the Brehm–Wintner–Conley dimension $\operatorname{bwc}(x)=\operatorname{rank}(S)$ (an integer invariant that stratifies central configurations and measures vertical degeneracy), together with the Plücker–BWC coordinates attached to it, we prove that each stratum of central configurations with fixed dimension and Brehm–Wintner–Conley dimension admits a base-point-free map into a Veronese variety, factoring through a Grassmannian invariant; on the Dziobek stratum this recovers the Dziobek–Veronese geometry previously introduced by the author. We further describe universal determinantal relations satisfied by the minors of $S$ and expand explicitly the resulting systems for the planar five- and six-body problems. We also interpret the mass-weighted entries of $S$ as an equilibrium stress: under the MacMillan–Bartky sign condition a strictly convex central configuration underlies a cable–strut tensegrity, and a theorem of Connelly then forces $S$ to be negative semidefinite with nullity three, so that vertical degeneracy cannot occur in this regime.

The associated, ideal-preserving, and locally associated properties of subrings, first formally defined in 2024, give a way of understanding the multiplicative structure of a subring given information about the larger ring. In this paper, we establish notation and preliminary results on a generalized type of polynomial and power series rings that is often used in the construction of counterexamples in the field of commutative algebra. We then provide sets of necessary and sufficient conditions under which such a polynomial or power series ring may be (locally) associated in a larger such ring. This allows for the production of several informative examples, as well as a better understanding of the circumstances under which the ring of formal power series $R[[x]]$ over an order $R$ in a number field may be half-factorial.

We exhibit an infinite family of disconnected multigraded Hilbert schemes on the biprojective space $\mathbb{P}^2_{\mathbb{k}} \times_\mathbb{k} \mathbb{P}^1_{\mathbb{k}}$. More precisely, for any integer $a \ge 2$ and $p_a(z_1,z_2) = 2az_1+az_2+3a-2a^2 \in \mathbb{Q}[z_1,z_2]$, the multigraded Hilbert scheme ${\rm Hilb}_{p_a}(\mathbb{P}^2_{\mathbb{k}}\times_\mathbb{k} \mathbb{P}^1_{\mathbb{k}})$ is disconnected. As a consequence, there exist infinitely many disconnected Haiman-Sturmfels multigraded Hilbert schemes even for a standard bigrading on a polynomial ring in only five variables.

We give an explicit counterexample in five variables to the Generalized Vanishing Conjecture. The construction is motivated by the recent counterexample to the Mathieu conjecture for SU(2). In the polynomial ring C[a,b,c,d,t], set P = (t+c)(ad+bt), Q = c, and let {\Lambda} = d/dt (d/da d/dd - d/db d/dc). We prove that {\Lambda}^m(P^m) = 0 for every m >= 1, whereas for every m >= 2, {\Lambda}^m(QP^m) = (-1)^m (m!)^2 (m+1)! t, which is nonzero. Thus the Generalized Vanishing Conjecture fails in dimension 5.

Matsumoto's algorithm for computing the radical of a polynomial ideal is generalized to the parametric setting within the framework of symbolic computation. The main tool is a comprehensive Gröbner system over a finite field, also known as a parametric Gröbner basis. As a result, an algorithm for computing the radical of a parametric ideal over a finite field is proposed.

Let the symmetric group $S_n$ act on the space of $n \times r$ real matrices by permuting rows, so orbits are multisets of $n$ points in $\mathbb{R}^r$. The basic $r$-symmetric tropical polynomials form a family of $\binom{n+r}{r}-1$ nonconstant invariants of degree at most $n$ that separates orbits and embeds the orbit space bi-Lipschitzly. We prove that this family generates the semifield of all $r$-symmetric tropical rational functions, answering a question raised in [J. Pure Appl. Algebra 223 (2019) 72-85]. Derksen showed that the invariant semifield of any permutation group $G \le S_N$ is generated in degree at most $N p_1 \cdots p_{|G|}$ ($p_i$ the $i$th prime), which for the row action is $nr p_1 \cdots p_{n!}$; the present result replaces this by generators of degree at most $n$. The generating expression is a finite minimum over the ways of re-assembling a multiset from its sorted columns, with penalties from the basic values that, via the bi-Lipschitz inequality, dominate a wrong re-assembly. The same penalties describe the image of the basic coordinate map as the zero set of a single tropical rational function and yield an expression algorithm. Subfamilies of the basic family containing the single-column values generate if and only if they separate. For any permutation group $G \le S_N$ the same mechanism generates the invariant semifield in degree at most $\max\{N, \binom{N}{2}\}$, a quadratic bound independent of the group order; combined with a genericity theorem of Cahill, Iverson, Mixon, and Packer, it yields $2N+1$ invariant tropical polynomials that separate orbits and $3N$ that generate, with at least $N$ necessary for each task. The quadratic bound is optimal: every $A_N$-invariant tropical polynomial of degree less than $\binom{N}{2}$ is $S_N$-invariant, so every separating family for the alternating group $A_N$ contains a member of degree at least $\binom{N}{2}$.

A graphic arrangement $\A_G$ associated with a simple graph $G$ is a classical and well-studied object in the theory of hyperplane arrangements. In this note, we show that, for a connected graph $G$, a slight modification of the logarithmic vector field $D(\A_G)$ of $\A_G$ is isomorphic to the face ring of a certain simplicial poset. This allows us to give formulas for several algebraic invariants of $D(\A_G)$, such as its Hilbert series, local cohomology, projective dimension, and Castelnuovo–Mumford regularity, in terms of combinatorial and topological information about the corresponding simplicial poset. As a by-product, we also give an explicit vector space basis of $D(\A_G)$.