Initial Ideals Research Articles

Blowup algebras of determinantal ideals in prime characteristic

AbstractWe study when blowup algebras are ‐split or strongly ‐regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals of Pfaffians of a skew‐symmetric matrix. We use these results to obtain bounds on the degrees of the defining equations for these algebras. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their depth stabilizes. Finally, we show that, for determinantal ideals, there exists a monomial order for which taking initial ideals commutes with taking symbolic powers. To obtain these results, we develop the notion of ‐split filtrations and symbolic ‐split ideals.

Cohen-Macaulay property of binomial edge ideals with girth of graphs

Conca and Varbaro (2020) [7] showed the equality of depth of a graded ideal and its initial ideal in a polynomial ring when the initial ideal is square-free. In this paper, we give some beautiful applications of this fact in the study of Cohen-Macaulay binomial edge ideals. We prove that for the characterization of Cohen-Macaulay binomial edge ideals, it is enough to consider only “biconnected graphs with some whisker attached” and this is done by investigating the initial ideals. We give several necessary conditions for a binomial edge ideal to be Cohen-Macaulay in terms of smaller graphs. Also, under a hypothesis, we give a sufficient condition for Cohen-Macaulayness of binomial edge ideals in terms of blocks of graphs. Moreover, we show that a graph with Cohen-Macaulay binomial edge ideal has girth less than 5 or equal to infinity.

Following Schubert varieties under Feigin’s degeneration of the flag variety

We study the effect of Feigin’s flat degeneration of the type A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {A}$$\\end{document} flag variety on the defining ideals of its Schubert varieties. In particular, we describe two classes of Schubert varieties which stay irreducible under the degenerations and in several cases we are able to encode reducibility of the degenerations in terms of symmetric group combinatorics. As a side result, we obtain an identification of some degenerate Schubert varieties (i.e. the vanishing sets of initial ideals of the ideals of Schubert varieties with respect to Feigin’s Gröbner degeneration) with Richardson varieties in higher rank partial flag varieties.

Gröbner bases, symmetric matrices, and type C Kazhdan–Lusztig varieties

AbstractWe study a class of combinatorially defined polynomial ideals that are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the symmetric Schubert determinantal ideals of A. Fink, J. Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a Kazhdan–Lusztig ideal of A. Woo and A. Yong; that is, it is the scheme‐theoretic defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The Kazhdan–Lusztig ideals that arise are exactly those where the opposite cell is 123‐avoiding. Our main results include Gröbner bases for these ideals, prime decompositions of their initial ideals (which are Stanley–Reisner ideals of subword complexes), and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams.

Generic generalized diagonal matrices

Generalized diagonal matrices are matrices that have two ladders of entries that are zero in the upper right and bottom left corners. The minors of generic generalized diagonal matrices have square-free initial ideals and the associated Stanley-Reisner complex can be viewed as a higher order complex of a certain poset. With this description, we characterize the configuration of ladders that yield Cohen-Macaulay ideals. In the special case where both ladders are triangles, we show that the corresponding complex is vertex decomposable.

Probing the predictive validity of ideal partner preferences for future partner traits and relationship outcomes across 13 years

The current study addresses the open question whether ideal partner preferences are linked to relationship decisions and relationship outcomes. Using a longitudinal design across 13 years, we investigated whether partner preferences are associated with perceived characteristics of actual partners (i.e. ideal-trait correlation) and whether a closer match between ideals and perceptions of a partner’s traits is associated with better relationship outcomes (i.e. ideal partner preference-matching effects). A community sample of 178 participants (90 women) reported their ideal partner preferences in 2006 (mean age at T2 M = 45.7 years, SD = 7.2). In 2019, they reported their relationship histories since then, providing ratings of 322 relationships. We found a positive association between participants’ initial ideals and partner trait perceptions. This ideal-trait correlation was stronger with current ideals, consistent with the possibility of preference adjustment towards the partner. The match between ideals and perceived partner traits was operationalised using different metrics. A closer match was associated with higher relationship commitment across all metrics, while for relationship quality, the link was not apparent for the corrected pattern metric. Evidence of matching effects for relationship length was mixed and largely absent for break-up initiation. Implications for the ideal partner preference literature are discussed.

Adjacency for scattering amplitudes from the Gröbner fan

Scattering amplitudes in planar N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory exhibit singularities which reflect various aspects of the cluster algebras associated to the Grassmannians Gr(4, n) and their tropical counterparts. Here we investigate the potential origins of such structures and examine the extent to which they can be recovered from the Gröbner structure of the underlying Plücker ideals, focussing on the Grassmannians corresponding to finite cluster algebras.Starting from the Plücker ideal, we describe how the polynomial cluster variables are encoded in non-prime initial ideals associated to certain maximal cones of the positive tropical fan. Following [1] we show that extending the Plücker ideal by such variables leads to a Gröbner fan with a single maximal Gröbner cone spanned by the positive tropical rays. The associated initial ideal encodes the compatibility relations among the full set of cluster variables. Thus we find that the Gröbner structure naturally encodes both the symbol alphabet and the cluster adjacency relations exhibited by scattering amplitudes without invoking the cluster algebra at all.As a potential application of these ideas we then examine the kinematic ideal associated to non-dual conformal massless scattering written in terms of spinor helicity variables. For five-particle scattering we find that the ideal can be identified with the Plücker ideal for Gr(3, 6) and the corresponding tropical fan contains a number of non-prime ideals which encode all additional letters of the two-loop pentagon function alphabet present in various calculations of massless five-point finite remainders.

Universally free numerical semigroups

A numerical semigroup is said to be universally free if it is free for any possible arrangement of its minimal generating set. In this work, we establish that toric ideals associated with universally free numerical semigroups can be generated by their set of circuits. Additionally, we provide a characterization of universally free numerical semigroups in terms of Gröbner bases. Specifically, a numerical semigroup is universally free if and only if all initial ideals of its corresponding toric ideal are complete intersections. Furthermore, we establish several equalities among the toric bases of a universally free numerical semigroup.We provide a complete characterization of 3-generated universally free numerical semigroups in terms of their minimal generating sets, and by proving the equality of certain toric bases. We compute exactly all the toric bases of a toric ideal defined by a 3-generated universally free numerical semigroup. Notably, we answer some questions posed by Tatakis and Thoma by demonstrating that toric ideals defined by 3-generated universally free numerical semigroups have a set of circuits and a universal Gröbner basis of size 3, while the universal Markov basis and the Graver basis can be arbitrarily large. We present partial results and propose several conjectures regarding universally free numerical semigroups with more than three generators.

The h-vectors of the edge rings of a special family of graphs

The h-vectors of homogeneous rings are one of the most important invariants that often reflect ring-theoretic properties. On the other hand, there are a few examples of edge rings of graphs whose h-vectors are explicitly computed. In this paper, we compute the h-vector of a special family of graphs, by using the technique of initial ideals and the associated simplicial complex.

Asymptotic invariants constructed from generic initial ideals

AbstractGeneric initial ideals (gins for short) were systematically introduced by Galligo in 1974 under the name of Grauert invariants since they appeared apparently first in works of Grauert and Hironaka. Ever since they have been of interest in commutative algebra and indirectly in algebraic geometry. Recently, Mayes in a series of articles associated with gins of graded families of ideals geometric objects called limiting shapes. The construction resembles that of Okunkov bodies but there are some differences as well. This work is motivated by Mayes articles and explores the connections between gins, limiting shapes, and some asymptotic invariants of homogeneous ideals which are associated with the gins, for example, asymptotic regularity, Waldschmidt constant and some new invariants, which seem relevant from geometric point of view.

A new class of finitely generated polynomial subalgebras without finite SAGBI bases

The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gröbner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely generated initial algebras. The class contains a subalgebra for which the set of initial algebras is uncountable, as well as a subalgebra with finitely many distinct initial algebras.

Frobenius methods in combinatorics

We survey results produced from the interaction between methods in prime characteristic and combinatorial commutative algebra. We showcase results for edge ideals, toric varieties, Stanley-Reisner rings, and initial ideals that were proven via Frobenius. We also discuss results for monomial ideals obtained using Frobenius-like maps. Finally, we present results for F-pure rings that were inspired by work done for Stanley-Reisner rings.

State polytopes related to two classes of combinatorial neural codes

Combinatorial neural codes are collections of 0/1 vectors that are used to model the co-firing patterns of a set of place cells in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically drawn on the plane as a Venn diagram-like figure. A sufficient condition to do so is for the code to have a property called 2-inductively pierced. Gross, Obatake, and Youngs recently used toric algebra to show that a code on three neurons is 1-inductively pierced if and only if the toric ideal is trivial or generated by quadratics. No result is known for additional neurons in the same generality, part of the difficulty coming from the large number of codewords possible when additional neurons are used.In this article, we study two infinite classes of combinatorial neural codes in detail. For each code, we explicitly compute its universal Gröbner basis. This is done for the first class by recognizing that the codewords form a Lawrence-type matrix. With the second class, this is done by showing that the matrix is totally unimodular. These computations allow one to compute the state polytopes of the corresponding toric ideals, from which all distinct initial ideals may be computed efficiently. Moreover, we show that the state polytopes are combinatorially equivalent to well-known polytopes: the permutohedron and the stellohedron.

On robustness and related properties on toric ideals

A toric ideal is called robust if its universal Gröbner basis is a minimal set of generators, and is called generalized robust if its universal Gröbner basis equals its universal Markov basis (the union of all its minimal sets of binomial generators). Robust and generalized robust toric ideals are both interesting from both a commutative algebra and an algebraic statistics perspective. However, only a few nontrivial examples of such ideals are known. In this work, we study these properties for toric ideals of both graphs and numerical semigroups. For toric ideals of graphs, we characterize combinatorially the graphs giving rise to robust and to generalized robust toric ideals generated by quadratic binomials. As a by-product, we obtain families of Koszul rings. For toric ideals of numerical semigroups, we determine that one of its initial ideals is a complete intersection if and only if the semigroup belongs to the so-called family of free numerical semigroups. Hence, we characterize all complete intersection numerical semigroups which are minimally generated by one of its Gröbner basis and, as a consequence, all the Betti numbers of the toric ideal and its corresponding initial ideal coincide. Moreover, also for numerical semigroups, we prove that the ideal is generalized robust if and only if the semigroup has a unique Betti element and that there are only trivial examples of robust ideals. We finish the paper with some open questions.

On initials and the fundamental theorem of tropical partial differential algebraic geometry

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The fundamental theorem of Tropical Differential Algebraic Geometry and its extensions state that the support of power series solutions of systems of ordinary differential equations (with formal power series coefficients over an uncountable algebraically closed field of characteristic zero) can be obtained either, by solving a so-called tropicalized differential system, or by testing monomial-freeness of the associated initial ideals. Tropicalized differential equations work on a completely different algebraic structure which may help in theoretical and computational questions, particularly on the existence of solutions.We show here that both of these methods can be generalized to the case of systems of partial differential equations, this is, one can go either with the solution of tropicalized systems, or test monomial-freeness of the ideal generated by the initials when looking for supports of power series solutions of systems of differential equations, regardless the (finite) number of derivatives. The key are the vertex sets of Newton polytopes, upon which relies the definition of both tropical vanishing condition and the initial of a differential polynomial.

Gröbner geometry for skew-symmetric matrix Schubert varieties

Matrix Schubert varieties are the closures of the orbits of B×B acting on all n×n matrices, where B is the group of invertible lower triangular matrices. Extending work of Fulton, Knutson and Miller identified a Gröbner basis for the prime ideals of these varieties. They also showed that the corresponding initial ideals are Stanley-Reisner ideals of shellable simplicial complexes, and derived a related primary decomposition in terms of reduced pipe dreams. These results lead to a geometric proof of the Billey-Jockusch-Stanley formula for a Schubert polynomial, among many other applications. We define skew-symmetric matrix Schubert varieties to be the nonempty intersections of matrix Schubert varieties with the subspace of skew-symmetric matrices. In analogy with Knutson and Miller's work, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gröbner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams. We show that these initial ideals are likewise the Stanley-Reisner ideals of shellable simplicial complexes. As an application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials. Our methods differ from Knutson and Miller's and can be used to give new proofs of some of their results, as we explain at the end of this article.

Radical Generic Initial Ideals

In this paper, we survey the theory of Cartwright–Sturmfels ideals. These are mathbb {Z}^{n}-graded ideals, whose multigraded generic initial ideal is radical. Cartwright–Sturmfels ideals have surprising properties, mostly stemming from the fact that their Hilbert scheme only contains one Borel-fixed point. This has consequences, e.g., on their universal Gröbner bases and on the family of their initial ideals. In this paper, we discuss several known classes of Cartwright–Sturmfels ideals and we find a new one. Among determinantal ideals of same-size minors of a matrix of variables and Schubert determinantal ideals, we are able to characterize those that are Cartwright–Sturmfels.

Linear strands of initial ideals of determinantal facet ideals

We construct an explicit linear strand for the initial ideal of any determinantal facet ideal (DFI) with respect to any diagonal term order. We show that if the clique complex of Δ has no 1-nonfaces larger than a certain cardinality, then the Betti numbers of the linear strand of and its initial ideal coincide. This confirms a conjecture of Ene–Herzog–Hibi for closed graphs with at most 2 maximal cliques. Additionally, we show that the linear strand of the initial ideal of any DFI is supported on an induced subcomplex of the complex of boxes introduced by Nagel–Reiner.

Expecting the unexpected: Quantifying the persistence of unexpected hypersurfaces

The independence of imposed vanishing conditions is a foundational issue for a wide range of research in algebraic geometry. In this spirit, if X⊂Pn is a reduced subscheme, we say that X admits an unexpected hypersurface of degree t and multiplicity m if requiring multiplicity m at a general point P fails to impose the expected number of conditions on the linear system of hypersurfaces of degree t containing X. Conditions which either guarantee the occurrence of unexpected hypersurfaces, or which ensure that they cannot occur, are not well understood. Research to date has made surprising connections to root systems, hyperplane arrangements, and generic splitting types of vector bundles, among other diverse topics. In this paper we introduce new methods for studying unexpectedness, such as the use of generic initial ideals and partial elimination ideals to clarify when it can and when it cannot occur. We also exhibit algebraic and geometric properties of X which in some cases guarantee and in other cases preclude X having certain kinds of unexpectedness. In addition, we formulate a new way of quantifying unexpectedness (our AV sequence), which allows us to detect the extent to which unexpectedness persists as t increases but t−m remains constant. Finally, we study to what extent we can detect unexpectedness from the Hilbert function of X.

A short note on generic initial ideals

The definition of a generic initial ideal includes the assumption $x_1>x_2> \cdots >x_n$. A natural question is how generic initial ideals change when we permute the variables. In the article [1, §2], it is shown that the generic initial ideals are permuted in the same way when the variables in the monomial order are permuted. We give a different proof of this theorem. Along the way, we study the Zariski open sets which play an essential role in the definition of a generic initial ideal and also prove a result on how the Zariski open set changes after a permutation of the variables.