Hilbert Function Research Articles

Topological Data Analysis on Simple English Wikipedia Articles

Single-parameter persistent homology, a key tool in topological data analysis, has been widely applied to data problems along with statistical techniques that quantify the significance of the results.
 In contrast, statistical techniques for two-parameter persistence, while highly desirable for real-world applications, have scarcely been considered.
 We present three statistical approaches for comparing geometric data using two-parameter persistent homology; these approaches rely on the Hilbert function, matching distance, and barcodes obtained from two-parameter persistence modules computed from the point-cloud data.
 Our statistical methods are broadly applicable for analysis of geometric data indexed by a real-valued parameter.
 We apply these approaches to analyze high-dimensional point-cloud data obtained from Simple English Wikipedia articles.
 In particular, we show how our methods can be utilized to distinguish certain subsets of the Wikipedia data and to compare with random data.
 These results yield insights into the construction of null distributions and stability of our methods with respect to noisy data.

Hilbert functions of certain rings of invariants via representations of the symmetric groups (with an appendix by Dejan Govc)

In this paper we study rings of invariants arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form K[U]Γ where Γ is a product of general linear groups over a field K of characteristic zero, and U is a finite dimensional rational representation of Γ. We will calculate the Hilbert series of such rings using the representation theory of the symmetric groups and Schur-Weyl duality. We focus on the case where U=End(W⊕k) and Γ=GL(W) and on the case where U=End(V⊗W) and Γ=GL(V)×GL(W), though the methods introduced here can also be applied in more general framework. For the two aforementioned cases we calculate the Hilbert function of the ring of invariants in terms of Littlewood-Richardson and Kronecker coefficients. When the vector spaces are of dimension 2 we also give an explicit calculation of this Hilbert series, using Mathematica (see the appendix by Dejan Govc).

The Hilbert function of bigraded algebras in $k[\mathbb{P}^1 \times \mathbb{P}^1

We classify the Hilbert functions of bigraded algebras in k[x1,x2,y1,y2] by introducing a numerical function called a Ferrers function.

4-generated pseudo symmetric monomial curves with not Cohen–Macaulay tangent cones

In this article, standard bases of some toric ideals associated to 4-generated pseudo symmetric semigroups with not Cohen-Macaulay tangent cones at the origin are computed. As the tangent cones are not Cohen-Macaulay, non-decreasingness of the Hilbert function of the local ring was not guaranteed. Therefore, using these standard bases, Hilbert functions are explicitly computed as a step towards the characterization of Hilbert function. In addition, when the smallest integer satisfying $k(\alpha_2+1)<(k-1)\alpha_1+(k+1)\alpha_{21}+\alpha_3$ is $1$, it is proved that the Hilbert function of the local ring is non-decreasing.

Decomposition of local cohomology tables of modules with large E-depth

We introduce the notion of E-depth of graded modules over polynomial rings to measure the depth of certain Ext modules. First, we characterize graded modules over polynomial rings with (sufficiently) large E-depth as those modules whose (sufficiently) partial general initial submodules preserve the Hilbert function of local cohomology modules supported at the irrelevant maximal ideal , extending a result of Herzog and Sbarra on sequentially Cohen-Macaulay modules. Second, we describe the cone of local cohomology tables of modules with sufficiently high E-depth, building on previous work of the second author and Smirnov. Finally, we obtain a non-Artinian version of a socle-lemma proved by Kustin and Ulrich.

Dynamic interaction between tidal current and upstream discharge at the Kapuas River mouth

This paper explores the dynamic interaction between tidal current and upstream discharge at the Kapuas River mouth. Long term time series data of tidal current and river discharge obtained from software Windwave 05 and hydrograph data, respectively. Three prominent parameters, i.e., river discharge, tides, and tidal current were analyzed using the wavelet method, Hilbert function, and low pass filter to evaluate the tidal characteristics and phase difference of M2, M4, K1, O1. The relationship between the tidal current and water level shows a mimic standing wave with the phase difference around ∼ -180 – 900, indicating the tidal current leads the water level. The low-pass filter results indicate that the increasing river discharge hampers the tides and reduces the tidal current at the mouth. The Hilbert function result reveals the asymmetric tide and flood dominant flow which has the opposite direction with the seaward direction of the current. Moreover, the subtidal water level flows in a seaward direction. This happened because the net seaward current modulated by the river discharge. The currents at the mouth are affected significantly by the swell wave.

Steiner systems and configurations of points

The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System S(t, n, v) we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement.

The range of all regularities for polynomial ideals with a given Hilbert function

Let A denote any polynomial ring over a field K and I any homogeneous ideal of A. In this paper it is proven that, given an Hilbert function u, the set of the regularities of the homogeneous ideals I such that the K-algebra A/I has Hilbert function u is an interval of integers. This result is achieved by means of constructive arguments related to the minimal functions with a given Hilbert polynomial and a given regularity.

Deformations of Calabi-Yau manifolds in Fano toric varieties

In this article, we investigate deformations of a Calabi-Yau manifold Z in a toric variety F, possibly not smooth. In particular, we prove that the forgetful morphism from the Hilbert functor HZF of infinitesimal deformations of Z in F to the functor of infinitesimal deformations of Z is smooth. This implies the smoothness of HZF at the corresponding point in the Hilbert scheme. Moreover, we give some examples and include some computations on the Hodge numbers of Calabi-Yau manifolds in Fano toric varieties.

Combinatorial decompositions for monomial ideals

It is well known that Riquier introduced the notion of multiplicative variables applied to order ideals to represent initial conditions of partial differential equations as series. On the other hands, Janet introduced the concept of involutive division. Following Riquier and Janet, we focus on the following problem.Suppose one needs to compute a monomial ideal generated in some degree D and its escalier, knowing the Hilbert function and some monomials which must belong to the ideal/escalier. We give combinatorial tools to answer this question. First we define combinatorial decompositions of sets of terms, showing the criteria to decompose the set T≥D of the terms in degree greater or equal than D, into disjoint subsets called cones. This is done by assigning to each term t of degree D some multiplicative variables. The cone of t is then the set formed by t and all its multiples obtained multiplying it by all possible products of powers of multiplicative variables. Then, supposed one has selected a decomposition, we deal with the escalier/ideal partition problem, namely we study the rules which force a term to be in the ideal (or in the escalier) provided that another term does, so that both the ideal and the escalier turn out to be decomposed in disjoint cones by the decomposition set on the whole T≥D.

Poset embeddings of Hilbert functions for two hypersurface rings

Let R=𝕂[a,b,c,d]∕(ad−bc) or 𝕂[a,b,c]∕(ac−b2), where 𝕂 is a field of arbitrary characteristic and a,b,c,d are indeterminates. In this paper we will show that there is an order preserving embedding from the poset of Hilbert functions of homogeneous ideals of R to the poset of homogeneous ideals of R. We also obtain related results for Betti numbers when the characteristic of 𝕂 is 0.

Instantons and Hilbert functions

We study superpotentials from worldsheet instantons in heterotic Calabi-Yau compactifications for vector bundles constructed from line bundle sums, monads and extensions. Within a certain class of manifolds and for certain second homology classes, we derive simple necessary conditions for a non-vanishing instanton superpotential. These show that non-vanishing instanton superpotentials are rare and require a specific pattern for the bundle construction. For the class of monad and extension bundles with this pattern, we derive a sufficient criterion for non-vanishing instanton superpotentials based on an affine Hilbert function. This criterion shows that a non-zero instanton superpotential is common within this class. The criterion can be checked using commutative algebra methods only and depends on the topological data defining the Calabi-Yau X and the vector bundle V.

On the configurations of codimension two linear subspaces of ℙN with the Waldschmidt constant less than 2

The purpose of this note is to generalize a result of [M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Symbolic powers of planar point configurations II, J. Pure Appl. Alg. 220 (2016) 2001–2016] to higher-dimensional projective spaces and classify all configurations of [Formula: see text]-planes [Formula: see text] in [Formula: see text] with the Waldschmidt constants less than two. We also determine some numerical and algebraic invariants of the defining ideals [Formula: see text] of these classes of configurations, i.e. the resurgence, the minimal free resolution and the regularity of [Formula: see text], as well as the Hilbert function of [Formula: see text].

Symmetric decomposition of the associated graded algebra of an Artinian Gorenstein algebra

We study the symmetric subquotient decomposition of the associated graded algebras A⁎ of a non-homogeneous commutative Artinian Gorenstein (AG) algebra A. This decomposition arises from the stratification of A⁎ by a sequence of ideals A⁎=CA(0)⊃CA(1)⊃⋯ whose successive quotients Q(a)=C(a)/C(a+1) are reflexive A⁎ modules. These were introduced by the first author [47,48], developed in the Memoir [49], and have been used more recently by several groups, especially those interested in short Gorenstein algebras, and in the scheme length (cactus rank) of forms.For us a Gorenstein sequence is an integer sequence H occurring as the Hilbert function H=H(A) for an AG algebra A, that is not necessarily homogeneous. Such a Hilbert function H(A) is the sum of symmetric non-negative sequences HA(a)=H(QA(a)), each having center of symmetry (j−a)/2 where j is the socle degree of A: we call these the symmetry conditions, and the decomposition D(A)=(HA(0),HA(1),…) the symmetric decomposition of H(A) (Theorem 1.4). We here study which sequences may occur as the summands HA(a): in particular we construct in a systematic way examples of AG algebras A for which HA(a) can have interior zeroes, as HA(a)=(0,s,0,…,0,s,0). We also study the symmetric decomposition sets D(A), and in particular determine which sequences HA(a) can be non-zero when the dual generator is linear in a subset of the variables (Theorem 4.1).Several groups have studied “exotic summands” of the Macaulay dual generator F: these are summands that involve more successive variables than would be expected from the symmetric decomposition of the Hilbert function H(A). Studying these, we recall a normal form for the Macaulay dual generator of an AG algebra that has no “exotic” summands (Theorem 2.7). We apply this to Gorenstein algebras that are connected sums (Section 2.4).We give throughout many examples and counterexamples, and conclude with some open questions about symmetric decomposition.

On minimal decompositions of low rank symmetric tensors

In this paper, we study the set of decompositions of symmetric tensors of low rank of any dimension and of arbitrary order and the Waring loci of points which appear in a decomposition of the tensor. We describe a new and complete stratification of the set of symmetric tensors of rank less than 5 by studying Hilbert function and regularity of ideals of points contained in the apolar ideal. For each stratum, we describe a procedure to compute a minimal apolar set of points, exploiting the algebraic properties of the Waring loci.

A generalization of Kruskal–Katona’s theorem

Abstract Let K be a field, E the exterior algebra of a finite dimensional K-vector space, and F a finitely generated graded free E-module with homogeneous basis g 1, . . ., gr such that deg g 1 ≤ deg g 2 ≤ · · · ≤ deg gr . We characterize the Hilbert functions of graded E–modules of the type F/M, with M graded submodule of F. The existence of a unique lexicographic submodule of F with the same Hilbert function as M plays a crucial role.

Universal lex ideal approximations of extended Hilbert functions and Hamilton numbers

Let R(h) denote the polynomial ring in variables x1,…,xh over a specified field K. We consider all of these rings simultaneously, and in each use lexicographic (lex) monomial order with x1>⋯>xh. Given a fixed homogeneous ideal I in R(h), for each d there is unique lex ideal generated in degree at most d whose Hilbert function agrees with the Hilbert function of I up to degree d. When we consider IR(N) for N≥h, the set Bd(I,N) of minimal generators for this lex ideal in degree at most d may change, but Bd(I,N) is constant for all N≫0. We let Bd(I) denote the set of generators one obtains for all N≫0, and we let bd=bd(I) be its cardinality. The sequences b1,…,bd,… obtained in this way may grow very fast. Remarkably, even when I=(x12,x22), one obtains a very interesting sequence, 0,2,3,4,6,12,924,409620,…. This sequence is the same as Hd−1+1 for d≥2, where Hd is the dth Hamilton number. The Hamilton numbers were studied by Hamilton and by Hammond and Sylvester because of their occurrence in a counting problem connected with the use of Tschirnhaus transformations in manipulating polynomial equations.

On the Hilbert function of Gorenstein algebras of socle degree four

The first example of a non unimodal Gorenstein h-vector was given by Stanley, it is (1,13,12,13,1). In [14] the authors showed that Stanley's example is optimal, i.e., the vector (1,12,11,12,1) is not a Gorenstein h-vector. Our main result is a generalization of this result. We also give a simple proof of Stanley's conjecture on the asymptotic behavior of the Hilbert function in socle degree four. We present a conjecture about the asymptotic behavior of the Hilbert function for those algebras that are presented by quadrics.

Bundles on with vanishing lower cohomologies

AbstractWe study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We partition the moduli $\mathcal{M}$ according to the Hilbert function H and classify all possible Hilbert functions H of such bundles. For each H, we describe a stratification of $\mathcal{M}_H$ by quotients of rational varieties. We show that the closed strata form a graded lattice given by the Betti numbers.

Quasisymmetric functions from combinatorial Hopf monoids and Ehrhart Theory

We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class of complexes, called forbidden composition complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf algebras, Ehrhart theory, and commutative algebra. We also study various specializations of quasisymmetric functions.