Maximal Minors Research Articles

Linkage and F-Regularity of Determinantal Rings

Abstract In this paper, we prove that the generic link of a generic determinantal ring defined by maximal minors is strongly $F$-regular. In the process, we strengthen a result of Chardin and Ulrich in the graded setting. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that if the said complete intersection is defined by homogeneous elements and is $F$-rational, then in fact, its generic residual intersections are strongly $F$-regular in positive prime characteristic. Hochster and Huneke showed that determinantal rings are strongly $F$-regular; however, their proof is quite involved. Our techniques allow us to give a new and simple proof of the strong $F$-regularity of determinantal rings defined by maximal minors.

Rees algebras of sparse determinantal ideals

We determine the defining equations of the Rees algebra and of the special fiber ring of the ideal of maximal minors of a 2 × n 2\times n sparse matrix. We prove that their initial algebras are ladder determinantal rings. This allows us to show that the Rees algebra and the special fiber ring are Cohen-Macaulay domains, they are Koszul, they have rational singularities in characteristic zero and are F-rational in positive characteristic.

Indecomposable integrally closed modules of rank 3 over two-dimensional regular local rings

We characterise ideals in two-dimensional regular local rings that arise as ideals of maximal minors of indecomposable integrally closed modules of rank three.

Socle degrees for local cohomology modules of thickenings of maximal minors and sub-maximal Pfaffians

Let S S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I I be the determinantal ideal of maximal minors or P f Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S S -module structures of E x t S j ( S / I t , S ) Ext^j_S(S/I^t, S) and E x t S j ( S / P f t , S ) Ext^j_S(S/Pf^t, S) , from which we get the degrees of generators of these E x t Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H m j ( S / I t ) H^j_\mathfrak {m}(S/I^t) .

Sagbi combinatorics of maximal minors and a Sagbi algorithm

The maximal minors of a matrix of indeterminates are a universal Gröbner basis by a theorem of Bernstein, Sturmfels and Zelevinsky. On the other hand it is known that they are not always a universal Sagbi basis. By an experimental approach we discuss their behavior under varying monomial orders and their extensions to Sagbi bases. These experiments motivated a new implementation of the Sagbi algorithm which is organized in a Singular script and falls back on Normaliz for the combinatorial computations. In comparison to packages in the current standard distributions of Macaulay 2, version 1.21, and Singular, version 4.2.1 and a package intended for CoCoA 5.4.2, it extends the range of computability by at least one order of magnitude.

A representation-theoretic interpretation of positroid classes

A positroid is the matroid of a real matrix with nonnegative maximal minors, a positroid variety is the closure of the locus of points in a complex Grassmannian whose matroid is a fixed positroid, and a positroid class is the cohomology class Poincaré dual to a positroid variety. We define a family of representations of general linear groups whose characters are symmetric polynomials representing positroid classes. These representations are certain diagram Schur modules in the sense of James and Peel. This gives a new algebraic interpretation of the Schubert structure constants for the product of a Schubert polynomial and Schur polynomial, and of the 3-point Gromov-Witten invariants for Grassmannians, proving a conjecture of Postnikov. As a byproduct, we obtain an effective algorithm for decomposing positroid classes into Schubert classes.

Gröbner bases and critical values: The asymptotic combinatorics of determinantal systems

Determinantal polynomial systems are those involving maximal minors of some given matrix. An important situation where these arise is the computation of the critical values of a polynomial map restricted to an algebraic set. This leads directly to a strategy for, among other problems, polynomial optimisation.Computing Gröbner bases is a classical method for solving polynomial systems in general. For practical computations, this consists of two main stages. First, a Gröbner basis is computed with respect to a DRL (degree reverse lexicographic) ordering. Then, a change of ordering algorithm, such as Sparse-FGLM, designed by Faugère and Mou, is used to find a Gröbner basis of the same system but with respect to a lexicographic ordering. The complexity of this latter step, in terms of the number of arithmetic operations in the ground field, is O(mD2), where D is the degree of the ideal generated by the input and m is the number of non-trivial columns of a certain D×D matrix.While asymptotic estimates are known for m in the case of generic polynomial systems, thus far, the complexity of Sparse-FGLM was unknown for the class of determinantal systems.By assuming Fröberg's conjecture, thus ensuring that the Hilbert series of generic determinantal ideals have the necessary structure, we expand the work of Moreno-Socías by detailing the structure of the DRL staircase in the determinantal setting. Then we study the asymptotics of the quantity m by relating it to the coefficients of these Hilbert series. Consequently, we arrive at a new bound on the complexity of the Sparse-FGLM algorithm for generic determinantal systems and, in particular, for generic critical point systems.We consider the ideal inside the polynomial ring K[x1,…,xn], where K is some infinite field, generated by p generic polynomials of degree d and the maximal minors of a p×(n−1) polynomial matrix with generic entries of degree d−1. Then, in this setting, for the case d=2 and for n≫p we establish an exact formula for m in terms of n and p. Moreover, for d≥3, we give a tight asymptotic formula, as n→∞, for m in terms of n,p and d.

Determinantal facet ideals for smaller minors

A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic $$n\times m$$ matrix where $$n\le m$$ indexed by the facets of a simplicial complex $$\Delta $$ . We consider the more general notion of an r-DFI, which is generated by a subset of r-minors of a generic matrix indexed by the facets of $$\Delta $$ for some $$1\le r\le n$$ . We define and study so-called lcm-closed and unit interval r-DFIs, and show that the minors parametrized by the facets of $$\Delta $$ form a reduced Gröbner basis with respect to any term order for an lcm-closed r-DFI. We also see that being lcm-closed generalizes conditions previously introduced in the literature, and conjecture that in the case $$r=n$$ , lcm-closedness is necessary for being a Gröbner basis. We also give conditions on the maximal cliques of $$\Delta $$ ensuring that lcm-closed and unit interval r-DFIs are Cohen-Macaulay. Finally, we conclude with a variant of a conjecture of Ene, Herzog, and Hibi on the Betti numbers of certain types of r-DFIs, and provide a proof of this conjecture for Cohen-Macaulay unit interval DFIs.

Kasteleyn Theorem, Geometric Signatures and KP-II Divisors on Planar Bipartite Networks in the Disk

Maximal minors of Kasteleyn sign matrices on planar bipartite graphs in the disk count dimer configurations with prescribed boundary conditions, and the weighted version of such matrices provides a natural parametrization of the totally non–negative part of real Grassmannians (Postnikov et al. J. Algebr. Combin. 30(2), 173–191, 2009; Lam J. Lond. Math. Soc. (2) 92(3), 633–656, 2015; Lam 2016; Speyer 2016; Affolter et al. 2019). In this paper we provide a geometric interpretation of such variant of Kasteleyn theorem: a signature is Kasteleyn if and only if it is geometric in the sense of Abenda and Grinevich (2019). We apply this geometric characterization to explicitly solve the associated system of relations and provide a new proof that the parametrization of positroid cells induced by Kasteleyn weighted matrices coincides with that of Postnikov boundary measurement map. Finally we use Kasteleyn system of relations to associate algebraic geometric data to KP multi-soliton solutions. Indeed the KP wave function solves such system of relations at the nodes of the spectral curve if the dual graph of the latter represents the soliton data. Therefore the construction of the divisor is automatically invariant, and finally it coincides with that in Abenda and Grinevich (Sel. Math. New Ser. 25(3), 43, 2019; Abenda and Grinevich 2020) for the present class of graphs.

A determinantal formula for circuits of integer lattices

Abstract Let L be a not necessarily saturated lattice in ℤ n with a defining matrix B. We explicitly compute the set of circuits of L in terms of maximal minors of B. This has a variety of applications from toric to tropical geometry, from Gröbner to Graver bases, and from linear to binomial ideals.

Trimming complexes and applications to resolutions of determinantal facet ideals

We produce a family of complexes called trimming complexes and explore its applications. We demonstrate how trimming complexes can be used to deduce the Betti table for the minimal free resolution of the ideal generated by certain subsets of a generating set for an arbitrary ideal I. In particular, we compute the Betti table of the ideal obtained by removing an arbitrary generator from the ideal of submaximal pfaffians of a generic skew symmetric matrix M. We also explicitly compute the Betti table for the ideal generated by certain subsets of the generating set of the ideal of maximal minors of a generic n × m matrix. Such ideals are a subset of a class of ideals called determinantal facet ideals, whose higher degree Betti numbers had not previously been computed.

The module of logarithmic derivations of a generic determinantal ideal

An important problem in algebra and related fields (such as algebraic and complex analytic geometry) is to find an explicit, well-structured, minimal set of generators for the module of logarithmic derivations of classes of homogeneous ideals in polynomial rings. In this note we settle the case of the ideal P ⊂ R = K [ { X i , j } ] P\subset R=K[\{X_{i,j}\}] generated by the maximal minors of an ( n + 1 ) × n (n+1)\times n generic matrix ( X i , j ) (X_{i,j}) over an arbitrary field K K with n ≥ 2 n\geq 2 . We also characterize when the derivation module of R / P R/P is Ulrich, and we investigate this property if we replace R / P R/P by determinantal rings arising from simple degenerations of the generic case.

Critical loci in computer vision and matrices dropping rank in codimension one

Critical loci for projective reconstruction from three views in four dimensional projective space are defined by an ideal generated by maximal minors of suitable 4×3 matrices, N, of linear forms. Such loci are classified in this paper, in the case in which N drops rank in codimension one, giving rise to reducible varieties. This rests on the complete classification of matrices of size (n+1)×n for n≤3, which drop rank in codimension one. Instability of reconstruction near non-linear components of critical loci is explored experimentally. The classification of special matrices as above is also leveraged to study degenerate critical loci for suitable projections from P3.

On the Multidimensional Tarry Problem for a Cubic Polynomial

A new upper bound for the exponent of convergence of a special integral in the Tarry problem is obtained. The result is based on the representation of a special integral as a surface integral extended to the manifold of solutions of the system of equations of the Tarry problem. New estimates of the arising surface integrals reducing the estimation to the study of operators with discrete spectrum are obtained by using maximal minors.

Factorization for n-D Polynomial Matrices over Arbitrary Coefficient Field

Multivariate (n-D) polynomial matrix factorization is one of important research contents in multidimensional (n-D) systems, circuits, and signal processing. In this paper, several results on n-D polynomial matrices factorization over arbitrary coefficient fields are proved. Based on these results, generalizations of some results on general matrix factorization are obtained for given n-D polynomial matrices whose maximal order minors or lower order minors satisfy certain conditions. The proposed results fit for arbitrary coefficient field and have a wide range of application.

Vandermonde determinantal ideals

We show that the ideal generated by maximal minors (i.e., $k+1$-minors) of a $(k+1) \times n$ Vandermonde matrix is radical and Cohen-Macaulay. Note that this ideal is generated by all Specht polynomials with shape $(n-k,1, …,1)$.

Acyclicity of Schur complexes and torsion freeness of Schur modules

Let R be a Noetherian commutative ring and M a R-module with pdRM≤1 that has rank. Necessary and sufficient conditions were provided in [1] for an exterior power ∧kM to be torsion free. When M is an ideal of R similar necessary and sufficient conditions were provided in [2] for a symmetric power SkM to be torsion free. We extend these results to a broad class of Schur modules Lλ/μM. En route, for any map of finite free R modules ϕ:F→G we also study the general structure of the Schur complexes Lλ/μϕ, and provide necessary and sufficient conditions for the acyclicity of any given Lλ/μϕ by computing explicitly the radicals of the ideals of maximal minors of all its differentials.

Strong F-regularity and generating morphisms of local cohomology modules

We establish a criterion for the strong F-regularity of a (non-Gorenstein) Cohen–Macaulay reduced complete local ring of dimension at least 2 and prime characteristic p. We also describe an explicit generating morphism (in the sense of Lyubeznik) for the top local cohomology module with support in certain ideals arising from an n×(n−1) matrix X of indeterminates. For a perfect field k of characteristic p≥5, these results led us to derive a simple, new proof of the well-known fact that the generic determinantal ring over k given by the maximal minors of X is strongly F-regular.

Minor Prime Factorization for n-D Polynomial Matrices over Arbitrary Coefficient Field

In this paper, we investigate two classes of multivariate (n-D) polynomial matrices whose coefficient field is arbitrary and the greatest common divisor of maximal order minors satisfy certain condition. Two tractable criterions are presented for the existence of minor prime factorization, which can be realized by programming and complexity computations. On the theory and application, we shall obtain some new and interesting results, giving some constructive computational methods for carrying out the minor prime factorization.

Rigid ideals by deforming quadratic letterplace ideals

We compute the deformation space of quadratic letterplace ideals L(2,P) of finite posets P when its Hasse diagram is a rooted tree. These deformations are unobstructed. The deformed family has a polynomial ring as the base ring. The ideal J(2,P) defining the full family of deformations is a rigid ideal and we compute it explicitly. In simple example cases J(2,P) is the ideal of maximal minors of a generic matrix, the Pfaffians of a skew-symmetric matrix, and a ladder determinantal ideal.