Let p be a prime and let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field F of characteristic p. For positive integers r and s with $r \leq s$, the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes J_{s}$ has the form $J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _{r}}$. This decomposition determines a partition $\lambda (r,s,p)=(\lambda _1,\lambda _2,\ldots , \lambda _{r})$ of $r s$. Let $n_1, \ldots , n_k$ be the multiplicities of the distinct parts of the partition and set $c(r,s,p)=(n_1,\ldots ,n_k)$. Then $c(r,s,p)$ is a composition of r. We present a new bottom-up algorithm for computing $c(r,s,p)$ and $\lambda (r,s,p)$ directly from the base-p expansions for r and s.

It is well-known that an element of a commutative ring with identity is nilpotent if, and only if , it lies in every prime ideal of the ring. A modification of this fact is amenable to a very simple proof mining analysis. We formulate a quantitative version of this modification and obtain an explicit bound. We present an application. This proof mining analysis is the leitmotif for some comments and observations on the methodology of computational extraction. In particular, we emphasize that the formulation of quantitative versions of ordinary mathematical theorems is of independent interest from proof mining metatheorems.

We calculate the integral equivariant cohomology, in terms of generators and relations, of locally standard torus orbifolds whose odd degree ordinary cohomology vanishes. We begin by studying GKM-orbifolds, which are more general, before specializing to half-dimensional torus actions.

In this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$ in $\Bbbk [x_1, \ldots , x_n]$ with $u_{ii} = 0, \ i\in \{ 1, \ldots , n\}$. We prove that the monomial curves in that family are set-theoretic complete intersections. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Frobenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.

Let I be a zero-dimensional ideal in the polynomial ring $K[x_1,\ldots ,x_n]$ over a field K. We give a bound for the number of roots of I in $K^n$ counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.

We relate the analytic spread of a module expressed as the direct sum of two submodules with the analytic spread of its components. We also study a class of submodules whose integral closure can be expressed in terms of the integral closure of its row ideals, and therefore can be obtained by means of a simple computer algebra procedure. In particular, we analyze a class of modules, not necessarily of maximal rank, whose integral closure is determined by the family of Newton polyhedra of their row ideals.

The (tree) amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is a certain subset of the Grassmannian introduced by Arkani-Hamed and Trnka in 2013 in order to study scattering amplitudes in $N=4$ supersymmetric Yang–Mills theory. Confirming a conjecture of the first author, we show that when $m$ is even, a collection of affine permutations yields a triangulation of $\mathcal {A}_{n,k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of $\mathcal {A}_{n,n-m-k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(n-k,n)$. We prove this duality using the twist map of Marsh and Scott. We also show that this map preserves the canonical differential forms associated with the corresponding positroid cells, and hence obtain a parity duality for amplituhedron differential forms.

We exhibit a set of generating relations for the modular invariant ring of a vector and a covector for the two-dimensional general linear group over a finite field.

Unfortunately, there is a mistake in [PS, Lemma 3.10] which invalidates [PS, Theorem 3.12]. We show that the theorem still holds if the ring is assumed to be Gorenstein.

We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with signature $(0,1,+,\cdot )$. In the characteristic zero case, the claim implies that the full theory is undecidable, for rings of the referred type. This extends a series of results by Raphael Robinson, holding for certain polynomial integral domains, to a more general class.

We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: an $\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$-cluster variety $\widehat {\mathcal {X} }$ to the toric variety associated to its g-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal {X}$-varieties encoded by $\operatorname {Star}(\tau )$ for each cone $\tau$ of the $\mathbf {g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal {A}_{\mathrm {prin}}$ of Gross, Hacking, Keel and Kontsevich [Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608], and the fibers cluster dual to $\mathcal {A} _t$. Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437–3527] with the Gross–Hacking–Keel–Kontsevich degeneration in the case of $\operatorname {Gr}_2(\mathbb {C} ^{5})$. Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

Assume that G is a graph with edge ideal $I(G)$ and star packing number $\alpha _2(G)$. We denote the sth symbolic power of $I(G)$ by $I(G)^{(s)}$. It is shown that the inequality $ \operatorname {\mathrm {depth}} S/(I(G)^{(s)})\geq \alpha _2(G)-s+1$ is true for every chordal graph G and every integer $s\geq 1$. Moreover, it is proved that for any graph G, we have $ \operatorname {\mathrm {depth}} S/(I(G)^{(2)})\geq \alpha _2(G)-1$.

Cluster algebras give rise to a class of Gorenstein rings which enjoy a large amount of symmetry. Concentrating on the rank 2 cases, we show how cluster varieties can be used to construct many interesting projective algebraic varieties. Our main application is then to construct hundreds of families of Fano 3-folds in codimensions 4 and 5. In particular, for Fano 3-folds in codimension 4 we construct at least one family for 187 of the 206 possible Hilbert polynomials contained in the Graded Ring Database.

We consider frieze sequences corresponding to sequences of cluster mutations for affine D- and E-type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient relations found by Keller and Scherotzke. Viewing the frieze sequence as a discrete dynamical system, we reduce it to a symplectic map on a lower dimensional space and prove Liouville integrability of the latter.

We consider maximal non-l-intertwining collections, which are a higher-dimensional version of the maximal non-crossing collections which give clusters of Plücker coordinates in the Grassmannian coordinate ring, as described by Scott. We extend a method of Scott for producing such collections, which are related to tensor products of higher Auslander algebras of type A. We show that a higher preprojective algebra of the tensor product of two d-representation-finite algebras has a d-precluster-tilting subcategory. Finally, we relate mutations of these collections to a form of tilting for these algebras.

The purpose of this paper is, as part of the stratification of Cohen–Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times _T S$ of Cohen–Macaulay local rings R, S of the same dimension $d>0$ over a regular local ring T with $\dim T=d-1$ is an almost Gorenstein ring if and only if so are R and S. In addition, the other generalizations of Gorenstein properties are also explored.

In representation theory, commutative algebra and algebraic geometry, it is an important problem to understand when the triangulated category $\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)=\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ admits a tilting (respectively, silting) object for a $\mathbb{Z}$-graded commutative Gorenstein ring $R=\bigoplus _{i\geqslant 0}R_{i}$. Here $\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)$ is the singularity category, and $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ is the stable category of $\mathbb{Z}$-graded Cohen–Macaulay (CM) $R$-modules, which are locally free at all nonmaximal prime ideals of $R$.

In this paper, we give a complete answer to this problem in the case where $\dim R=1$ and $R_{0}$ is a field. We prove that $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ always admits a silting object, and that $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ admits a tilting object if and only if either $R$ is regular or the $a$-invariant of $R$ is nonnegative. Our silting/tilting object will be given explicitly. We also show that if $R$ is reduced and nonregular, then its $a$-invariant is nonnegative and the above tilting object gives a full strong exceptional collection in $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R=\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}R$.

We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (for example, standard difference schemes) and difference equations in functions on words. On the universality side, we prove a version of strong Nullstellensatz for such difference equations under the assumption that the cardinality of the ground field is greater than the cardinality of the monoid and construct an example showing that this assumption cannot be omitted. On the undecidability side, we show that the following problems are undecidable:

We study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$-extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$-extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$-extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.