Hilbert Function Research Articles

Numerical implicitization

We present the $\textit{NumericalImplicitization}$ package for $\textit{Macaulay2}$, which allows for user-friendly computation of the invariants of the image of a polynomial map, such as dimension, degree, and Hilbert function values. This package relies on methods of numerical algebraic geometry, including homotopy continuation and monodromy.

On almost revlex ideals with Hilbert function of complete intersections

In this paper, we investigate the behavior of almost reverse lexicographic ideals with the Hilbert function of a complete intersection. More precisely, over a field K, we give a new constructive proof of the existence of the almost revlex ideal $$J\subset K[x_1,\ldots ,x_n]$$ , with the same Hilbert function as a complete intersection defined by n forms of degrees $$d_1\le \cdots \le d_n$$ . Properties of the reduction numbers for an almost revlex ideal have an important role in our inductive and constructive proof, which is different from the more general construction given by Pardue in 2010. We also detect several cases in which an almost revlex ideal having the same Hilbert function as a complete intersection corresponds to a singular point in a Hilbert scheme. This second result is the outcome of a more general study of lower bounds for the dimension of the tangent space to a Hilbert scheme at stable ideals, in terms of the number of minimal generators.

FPGA Implementation and Evaluation of an Approximate Hilbert Transform-Based Envelope Detector for Ultrasound Imaging Using the DSP Builder Development Tool.

In this paper, we present the FPGA implementation of an approximate Hilbert Transform-based envelope detector to compute the magnitude of the received ultrasound echo signals in real-time using a Model-based design flow. The proposed architecture exploits the negative odd-symmetry and interleaved zero-valued coefficients of a Hilbert Transform-based FIR filter to reduce hardware resource requirements and complexity. The hardware design is modeled using the DSP Builder development tool allowing the automatic generation of HDL algorithms directly from the Matlab/Simulink environment. The generated VHDL code was synthesized for an Intel Stratix IV FPGA and validated on a Terasic DE4-230 board. The accuracy and performance of the envelope detector are analyzed with real ultrasound phantom data for different filter orders, coefficient length and two filter design methods: Equiripple and Least-Squares. The normalized residual sum of squares (NRSS) and the normalized root mean square error (NRMSE) cost functions are used for comparison with the reference absolute value of the Matlab Hilbert function. The results demonstrate that the proposed method yields similar results to conventional envelope detection methods, while being simpler to implement and requiring lower computational cost.

Identifiability for a Class of Symmetric Tensors

We use methods of algebraic geometry to find new, effective methods for detecting the identifiability of symmetric tensors. In particular, for ternary symmetric tensors T of degree 7, we use the analysis of the Hilbert function of a finite projective set, and the Cayley–Bacharach property, to prove that, when the Kruskal’s ranks of a decomposition of T are maximal (a condition which holds outside a Zariski closed set of measure 0), then the tensor T is identifiable, i.e., the decomposition is unique, even if the rank lies beyond the range of application of both the Kruskal’s and the reshaped Kruskal’s criteria.

Vanishing Ideals Over Odd Cycles

Abstract Let K be a finite field. Let X * be a subset of the a ne space Kn , which is parameterized by odd cycles. In this paper we give an explicit Gröbner basis for the vanishing ideal, I(X *), of X *. We give an explicit formula for the regularity of I(X *) and finally if X * is parameterized by an odd cycle of length k, we show that the Hilbert function of the vanishing ideal of X * can be written as linear combination of Hilbert functions of degenerate torus.

Fat points, partial intersections and Hamming distance

We use two main techniques, namely, residuation and separators of points, to show that the Hilbert function of a certain fat point set supported on a grid complete intersection is the same as the Hilbert function of a reduced set of points called a partial intersection. As an application, we answer a question of Tohǎneanu and Van Tuyl which relates the minimum Hamming distance of a special linear code and the minimum socle degree of the associated fat point set.

Hilbert schemes, Verma modules and spectral functions of hyperbolic geometry with application to quantum invariants

In this paper we exploit Ruelle-type spectral functions and analyze the Verma module over Virasoro algebra, boson–fermion correspondence, the analytic torsion, the Chern–Simons and [Formula: see text] invariants, as well as the generating function associated to dimensions of the Hochschild homology of the crossed product [Formula: see text] ([Formula: see text] is the [Formula: see text]-Weyl algebra). After analyzing the Chern–Simons and [Formula: see text] invariants of Dirac operators by using irreducible [Formula: see text]-flat connections on locally symmetric manifolds of nonpositive section curvature, we describe the exponential action for the Chern–Simons theory.

A note on the asymptotics of the number of [formula omitted]-sequences of given length

We look at the number L(n) of O-sequences of length n. Recall that an O-sequence can be defined algebraically as the Hilbert function of a standard graded k-algebra, or combinatorially as the f-vector of a multicomplex. The sequence L(n) was first investigated in a recent paper by commutative algebraists Enkosky and Stone, inspired by Huneke. In this note, we significantly improve both of their upper and lower bounds, by means of a very short partition-theoretic argument. In particular, it turns out that, for suitable positive constants c1 and c2 and all n>2, ec1n≤L(n)≤ec2nlogn. It remains an open problem to determine an exact asymptotic estimate for L(n).

Tangential varieties of Segre–Veronese surfaces are never defective

We compute the dimensions of all the secant varieties to the tangential varieties of all Segre-Veronese surfaces. We exploit the typical approach of computing the Hilbert function of special 0-dimensional schemes on projective plane by using a new degeneration technique.

Smoothable Gorenstein Points Via Marked Schemes and Double-generic Initial Ideals

Over an infinite field K with we investigate smoothable Gorenstein K-points in a punctual Hilbert scheme and obtain the following results: (i) every K-point defined by local Gorenstein K-algebras with Hilbert function is smoothable (this is the only case non treated in the range considered by Iarrobino and Kanev in 1999; (ii) the Hilbert scheme has at least five irreducible components. As a byproduct of our study about we also find a new elementary component in We face the problem from a new point of view, that is based on properties of double-generic initial ideals and of marked schemes. The properties of marked schemes give us a simple method to compute the Zariski tangent space to a Hilbert scheme at a given K-point, which is very useful in this context. We also test our tools to find the already known result that K-points defined by local Gorenstein K-algebras with Hilbert function are smoothable. The problem that we consider is strictly related to the study of the irreducibility of the Gorenstein locus in a Hilbert scheme and, more generally, of the irreducibility of a Hilbert scheme, which is a very open question.

Sequences of quadratic transforms of a noetherian local domain

Let R be a noetherian local domain. The aim of this paper is to characterize the sequences (Ri)i≥0 of successive quadratic transforms of R=R0 such that S=⋃i≥0Ri is a valuation ring. These characterizations involve ideals of S supported on the exceptional divisors along the sequence. Furthermore, the characterizations are improved remarkably when the Hilbert functions of the rings (Ri)i≥0 stabilize along the sequence. We also give some examples to put our results in perspective.

Free extensions and Lefschetz properties, with an application to rings of relative coinvariants

Graded Artinian algebras can be regarded as algebraic analogues of cohomology rings (in even degrees) of compact topological manifolds. In this analogy, a free extension of a base ring with a fibre ring corresponds to a fibre bundle over a manifold. If the manifold is Kähler, then its cohomology ring satisfies the strong Lefschetz property, which means multiplication by a linear form has the largest possible Jordan type. In this paper, we study the behaviour of strong Lefschetz and Jordan type with respect to free extensions, using relative coinvariant rings of finite groups as prototypical models. We show that if V is a vector space and if the subgroup W of the general linear group Gl(V) is a non-modular finite reflection group and is a non-parabolic reflection subgroup, then the relative coinvariant ring cannot have a linear element of strong Lefschetz Jordan type. However, we give examples where these rings , some with non-unimodal Hilbert functions, nevertheless have (non-homogeneous) elements of strong Lefschetz Jordan type. Some of these examples are related to open combinatorial questions proposed and partially solved by G. Almkvist.

Secant varieties of the varieties of reducible hypersurfaces in [formula omitted

Given the space V=P(d+n−1n−1)−1 of forms of degree d in n variables, and given an integer ℓ>1 and a partition λ of d=d1+⋯+dr, it is in general an open problem to obtain the dimensions of the (ℓ−1)-secant varieties σℓ(Xn−1,λ) for the subvariety Xn−1,λ⊂V of hypersurfaces whose defining forms have a factorization into forms of degrees d1,…,dr. Modifying a method from intersection theory, we relate this problem to the study of the Weak Lefschetz Property for a class of graded algebras, based on which we give a conjectural formula for the dimension of σℓ(Xn−1,λ) for any choice of parameters n,ℓ and λ. This conjecture gives a unifying framework subsuming all known results. Moreover, we unconditionally prove the formula in many cases, considerably extending previous results, as a consequence of which we verify many special cases of previously posed conjectures for dimensions of secant varieties of Segre varieties. In the special case of a partition with two parts (i.e., r=2), we also relate this problem to a conjecture by Fröberg on the Hilbert function of an ideal generated by general forms.

Extremal behavior in sectional matrices

In this paper, we recall the object sectional matrix which encodes the Hilbert functions of successive hyperplane sections of a homogeneous ideal. We translate and/or reprove recent results in this language. Moreover, some new results are shown about their maximal growth, in particular, a new generalization of Gotzmann’s Persistence Theorem, the presence of a GCD for a truncation of the ideal, and applications to saturated ideals.

The initial ideal of generic sequences and Fröberg's Conjecture

Let K be an infinite field and let I=(f1,⋯,fr) be an ideal in the polynomial ring R=K[x1,⋯,xn] generated by generic forms of degrees d1,⋯,dr. A longstanding conjecture by Fröberg predicts the shape of the Hilbert function of R/I. In 2010 Pardue stated a conjecture on the initial ideal of n generic forms with respect to the deg-revlex order and he proved that it is equivalent to Fröberg's Conjecture. We study Pardue's Conjecture and we prove it under suitable conditions on the degrees of the forms. This yields a partial solution to Fröberg's Conjecture in the case r≤n+2 over an infinite field of any characteristic.

Proof of the Gorenstein Interval Conjecture in low socle degree

Roughly ten years ago, the following “Gorenstein Interval Conjecture” (GIC) was proposed: Whenever (1,h1,…,hi,…,he−i,…,he−1,1) and (1,h1,…,hi+α,…,he−i+α,…,he−1,1) are both Gorenstein Hilbert functions for some α≥2, then (1,h1,…,hi+β,…,he−i+β,…,he−1,1) is also Gorenstein, for all β=1,2,…,α−1. Since an explicit characterization of which Hilbert functions are Gorenstein is widely believed to be hopeless, the GIC, if true, would at least provide the existence of a strong, and very natural, structural property for such basic functions in commutative algebra. Before now, very little progress was made on the GIC.The main goal of this note is to prove the case e≤5, in arbitrary codimension. Our arguments will be in part constructive, and will combine several different tools of commutative algebra and classical algebraic geometry.

On the Cayley-Bacharach Property

The Cayley-Bacharach Property (CBP), which has been classically stated as a property of a finite set of points in an affine or projective space, is extended to arbitrary 0-dimensional affine algebras over arbitrary base fields. We present characterizations and explicit algorithms for checking the CBP directly, via the canonical module, and in combination with the property of being a locally Gorenstein ring. Moreover, we characterize strict Gorenstein rings by the CBP and the symmetry of their affine Hilbert function, as well as by the strict CBP and the last difference of their affine Hilbert function.

On $f$- and $h$-vectors of relative simplicial complexes

A relative simplicial complex is a collection of sets of the form $\Delta \setminus \Gamma$, where $\Gamma \subset \Delta$ are simplicial complexes. Relative complexes played key roles in recent advances in algebraic, geometric, and topological combinatorics but, in contrast to simplicial complexes, little is known about their general combinatorial structure. In this paper, we address a basic question in this direction and give a characterization of $f$-vectors of relative (multi)complexes on a ground set of fixed size. On the algebraic side, this yields a characterization of Hilbert functions of quotients of homogeneous ideals over polynomial rings with a fixed number of indeterminates. Moreover, we characterize $h$-vectors of fully Cohen--Macaulay relative complexes as well as $h$-vectors of Cohen--Macaulay relative complexes with minimal faces of given dimensions. The latter resolves a question of Bj\"orner.

Hilbert polynomials of the algebras of $SL_ 2$-invariants

We consider one of the fundamental problems of classical invariant theory, the research of Hilbert polynomials for an algebra of invariants of Lie group $SL_2$. Form of the Hilbert polynomials gives us important information about the structure of the algebra. Besides, the coefficients and the degree of the Hilbert polynomial play an important role in algebraic geometry. It is well known that the Hilbert function of the algebra $SL_n$-invariants is quasi-polynomial. The Cayley-Sylvester formula for calculation of values of the Hilbert function for algebra of covariants of binary $d$-form $\mathcal{C}_{d}= \mathbb{C}[V_d\oplus \mathbb{C}^2]_{SL_2}$ (here $V_d$ is the $d+1$-dimensional space of binary forms of degree $d$) was obtained by Sylvester. Then it was generalized to the algebra of joint invariants for $n$ binary forms. But the Cayley-Sylvester formula is not expressed in terms of polynomials.In our article we consider the problem of computing the Hilbert polynomials for the algebras of joint invariants and joint covariants of $n$ linear forms and $n$ quadratic forms. We express the Hilbert polynomials $\mathcal{H} \mathcal{I}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i, \mathcal{H}(\mathcal{C}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i,$ $\mathcal{H}(\mathcal{I}^{(n)}_2,i)=\dim(\mathcal{I}^{(n)}_2)_i, \mathcal{H}(\mathcal{C}^{(n)}_2,i)=\dim(\mathcal{C}^{(n)}_2)_i$ of those algebras in terms of quasi-polynomial. We also present them in the form of Narayana numbers and generalized hypergeometric series.

Indices of normalization of ideals

We derive numerical estimates controlling the intertwined properties of the normalization of an ideal and of the computational complexity of general processes for its construction. In [18], this goal was carried out for equimultiple ideals via the examination of Hilbert functions. Here we add to this picture, in an important case, how certain Hilbert functions provide a description of the locations of the generators of the normalization of ideals of dimension zero. We also present a rare instance of normalization of a class of homogeneous ideals by a single colon operation.