Hilbert Function Research Articles

Abstract Coherent State Transforms Over Homogeneous Spaces of Compact Groups

This paper presents theoretical aspects of a unified generalization for the abstract theory of coherent state/voice transforms over homogeneous spaces of compact groups using operator theory. Let G be a compact group and H be a closed subgroup of G. Let G/H be the left coset space of H in G and $$\mu $$ be the normalized G-invariant measure on G/H associated to the Weil’s formula with respect to the probability measures of G, H. Let $$(\pi ,\mathcal {H}_\pi )$$ be a continuous unitary representation of G with non-zero mean over H. In this article, we introduce the generalized notion of coherent state/voice transform associated to $$\pi $$ on the Hilbert function $$L^2(G/H,\mu )$$ . We then study basic analytic properties of these transforms.

Abstract relative Gabor transforms over canonical homogeneous spaces of semidirect product groups with Abelian normal factor

This paper introduces a unified approach to the abstract notion of relative Gabor transforms over canonical homogeneous spaces of semi-direct product groups with Abelian normal factor. Let [Formula: see text] be a locally compact group, [Formula: see text] be a locally compact Abelian (LCA) group, and [Formula: see text] be a continuous homomorphism. Let [Formula: see text] be the semi-direct product of [Formula: see text] and [Formula: see text] with respect to [Formula: see text], [Formula: see text] be the canonical homogeneous space of [Formula: see text], and [Formula: see text] be the canonical relatively invariant measure on [Formula: see text]. Then we present a unified harmonic analysis approach to the theoretical aspects of the notion of relative Gabor transform over the Hilbert function space [Formula: see text].

On semigroup rings with decreasing Hilbert function

Given a one-dimensional semigroup ring R=k[[S]], in this article we study the behaviour of the Hilbert function HR. By means of the notion of support of the elements in S, for some classes of semigroup rings we give conditions on the generators of S in order to have decreasing HR. When the embedding dimension v and the multiplicity e verify v+3≤e≤v+4, the decrease of HR gives an explicit description of the Apéry set of S. In particular for e=v+3, we prove that HR is non-decreasing if e≤12 and we classify the semigroups with e=13 and HR decreasing. Finally we deduce that HR is non-decreasing for every Gorenstein semigroup ring with e≤v+4. This fact is not true in general: through numerical duplication and some of the above results another recent paper shows the existence of infinitely many one-dimensional Gorenstein rings with decreasing Hilbert function.

Hilbert transform-based frequency response function estimation from run-up response of rotors

Transient operating conditions such as run-up and run-down are often seen in automotive and other machines with variable speed rotating elements. To estimate the frequency response function from swept sine responses (obtained during run-up), a new approach is proposed which utilizes Hilbert transform and stationary wavelet transform. This method is an alternate tool for conventional swept sine frequency response function estimators like tracking vectors and co-quad analysers. The output from this algorithm is validated using frequency response function obtained using exact frequency response function expression and the conventional Fourier transform-based frequency response function. Both MATLAB simulations and experiments have been carried out in this study to estimate the frequency response function. Modal parameters are extracted from the frequency response function obtained using Hilbert transform.

One-dimensional Gorenstein local rings with decreasing Hilbert function

In this paper we solve a problem posed by M.E. Rossi: Is the Hilbert function of a Gorenstein local ring of dimension one not decreasing? More precisely, for any integer h>1, h∉{14+22k,35+46k|k∈N}, we construct infinitely many one-dimensional Gorenstein local rings, included integral domains, reduced and non-reduced rings, whose Hilbert function decreases at level h; moreover, we prove that there are no bounds to the decrease of the Hilbert function. The key tools are numerical semigroup theory, especially some necessary conditions to obtain decreasing Hilbert functions found by the first and the third author, and a construction developed by V. Barucci, M. D'Anna and the second author, that gives a family of quotients of the Rees algebra. Many examples are included.

On the analysis of Hilbert module-valued functions on LCA groups

On the analysis of Hilbert module-valued functions on LCA groups

The Hilbert function of some Hadamard products

In this paper we address the Hadamard product of not necessarily generic linear varieties, looking in particular at its Hilbert function. We find that the Hilbert function of the Hadamard product $$X\star Y$$ of two varieties, with $$\dim (X), \dim (Y)\le 1$$ , is the product of the Hilbert functions of the original varieties X and Y. Moreover, the same result is obtained for generic linear varieties X and Y as a consequence of our showing that their Hadamard product is projectively equivalent to a Segre embedding.

On the score sheets of a round-robin football tournament

The set of (ordered) score sheets of a round-robin football tournament played between n teams together with the pointwise addition has the structure of an affine monoid. In this paper we study (using both theoretical and computational methods) the most important invariants of this monoid, namely the Hilbert basis, the multiplicity, the Hilbert series and the Hilbert function.

A structure theorem for most unions of complete intersections

Using the connections among almost complete intersection schemes, arithmetically Gorenstein schemes and schemes that are a union of complete intersections, we give a structure theorem for the arithmetically Cohen-Macaulay union of two complete intersections of codimension~2, of type $(d_1,e_1)$ and $(d_2,e_2)$ such that $\min \{d_1,e_1\}\ne \min \{d_2,e_2\}$. We apply the results for computing Hilbert functions and graded Betti numbers for such schemes.

Families of artinian and low dimensional determinantal rings

Let GradAlg(H) be the scheme parameterizing graded quotients of R=k[x0,…,xn] with Hilbert function H (it is a subscheme of the Hilbert scheme of Pn if we restrict to quotients of positive dimension, see definition below). A graded quotient A=R/I of codimension c is called standard determinantal if the ideal I can be generated by the t×t minors of a homogeneous t×(t+c−1) matrix (fij). Given integers a0≤a1≤...≤at+c−2 and b1≤...≤bt, we denote by Ws(b_;a_)⊂GradAlg(H) the stratum of determinantal rings where fij∈R are homogeneous of degrees aj−bi.In this paper we extend previous results on the dimension and codimension of Ws(b_;a_) in GradAlg(H) to artinian determinantal rings, and we show that GradAlg(H) is generically smooth along Ws(b_;a_) under some assumptions. For zero and one dimensional determinantal schemes we generalize earlier results on these questions. As a consequence we get that the general element of a component W of the Hilbert scheme of Pn is glicci provided W contains a standard determinantal scheme satisfying some conditions. We also show how certain ghost terms disappear under deformation while other ghost terms remain and are present in the minimal resolution of a general element of GradAlg(H).

Non-level semi-standard graded Cohen–Macaulay domain with h-vector (h0,h1,h2)

Let k be an algebraically closed field of characteristic 0, and A=⨁i∈NAi a Cohen–Macaulay graded domain with A0=k. If A is semi-standard graded (i.e., A is finitely generated as a k[A1]-module), it has the h-vector(h0,h1,…,hs), which encodes the Hilbert function of A. From now on, assume that s=2. It is known that if A is standard graded (i.e., A=k[A1]), then A is level. We will show that, in the semi-standard case, if A is not level, then h1+1 divides h2. Conversely, for any positive integers h and n, there is a non-level A with the h-vector (1,h,(h+1)n). Moreover, such examples can be constructed as Ehrhart rings (equivalently, normal toric rings).

Partially symmetric tensor rank: the description of the non-uniqueness case for low rank

We study the partially symmetric tensors T with low rank whose rank is not achieved by a unique set when the rank has an upper bound like , where is the minimum of the entries of the multidegree of T. For the proof, we use the Hilbert function of finite subsets of a product of projective spaces with respect to a Segre–Veronese embedding.

On Gröbner bases and Krull dimension of residue class rings of polynomial rings over integral domains

Given an ideal a in A[x1,…,xn] where A is a Noetherian integral domain, we propose an approach to compute the Krull dimension of A[x1,…,xn]/a, when the residue class ring is a free A-module. When A is a field, the Krull dimension of A[x1,…,xn]/a has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetherian rings. For a Noetherian integral domain A we introduce the notion of combinatorial dimension of A[x1,…,xn]/a and give a Gröbner basis method to compute it for residue class rings that have a free A-module representation w.r.t. a lexicographic ordering. For such A-algebras, we derive a relation between Krull dimension and combinatorial dimension of A[x1,…,xn]/a. An immediate application of this relation is that it gives a uniform method, the first of its kind, to compute the dimension of A[x1,…,xn]/a without having to consider individual properties of the ideal. For A-algebras that have a free A-module representation w.r.t. degree compatible monomial orderings, we introduce the concepts of Hilbert function, Hilbert series and Hilbert polynomials and show that Gröbner basis methods can be used to compute these quantities. We then proceed to show that the combinatorial dimension of such A-algebras is equal to the degree of the Hilbert polynomial. This enables us to extend the relation between Krull dimension and combinatorial dimension to A-algebras with a free A-module representation w.r.t. a degree compatible ordering as well.

Hilbert functions of Cox rings of del Pezzo surfaces

To study syzygies of the Cox rings of del Pezzo surfaces, we calculate important syzygetic invariants such as the Hilbert functions, the Green–Lazarsfeld indices, the projective dimensions, and the Castelnuovo–Mumford regularities. Using these computations as well as the natural multigrading structures by the Picard groups of del Pezzo surfaces and Weyl group actions on Picard lattices, we determine the Betti diagrams of the Cox rings of del Pezzo surfaces of degree at most four.

On the Real-Rootedness of the Veronese Construction for Rational Formal Power Series

We study real sequences $\{a_{n}\}_{n\in \mathbb{N}}$ that eventually agree with a polynomial. We show that if the numerator polynomial of its rational generating series is of degree $s$ and has only nonnegative coefficients, then the numerator polynomial of the subsequence $\{ a_{rn+i}\}_{n\in \mathbb{N}}$, $0\leq i<r$, has only nonpositive, real roots for all $r\geq s-i$. We apply our results to combinatorially positive valuations on polytopes and to Hilbert functions of Veronese submodules of graded Cohen-Macaulay algebras. In particular, we prove that the Ehrhart $h^\ast$-polynomial of the $r$-th dilate of a $d$-dimensional polytope has only distinct, negative, real roots if $r\geq \min \{s+1,d\}$. This proves a conjecture of Beck and Stapledon (2010).

Matroid configurations and symbolic powers of their ideals

Star configurations are certain unions of linear subspaces of projective space that have been studied extensively. We develop a framework for studying a substantial generalization, which we call matroid configurations, whose ideals generalize Stanley-Reisner ideals of matroids. Such a matroid configuration is a union of complete intersections of a fixed codimension. Relating these to the Stanley-Reisner ideals of matroids and using methods of liaison theory allows us, in particular, to describe the Hilbert function and minimal generators of the ideal of, what we call, a hypersurface configuration. We also establish that the symbolic powers of the ideal of any matroid configuration are Cohen-Macaulay. As applications, we study ideals coming from certain complete hypergraphs and ideals derived from tetrahedral curves. We also consider Waldschmidt constants and resurgences. In particular, we determine the resurgence of any star configuration and many hypersurface configurations. Previously, the only non-trivial cases for which the resurgence was known were certain monomial ideals and ideals of finite sets of points. Finally, we point out a connection to secant varieties of varieties of reducible forms.

On factorization invariants and Hilbert functions

Nonunique factorization in cancellative commutative semigroups is often studied using combinatorial factorization invariants, which assign to each semigroup element a quantity determined by the factorization structure. For numerical semigroups (additive subsemigroups of the natural numbers), several factorization invariants are known to admit predictable behavior for sufficiently large semigroup elements. In particular, the catenary degree and delta set invariants are both eventually periodic, and the omega-primality invariant is eventually quasilinear. In this paper, we demonstrate how each of these invariants is determined by Hilbert functions of graded modules. In doing so, we extend each of the aforementioned eventual behavior results to finitely generated semigroups, and provide a new framework through which to study factorization structures in this setting.

Square-integrability of multivariate metaplectic wave-packet representations

This paper presents a systematic study for harmonic analysis of metaplectic wave-packet representations on the Hilbert function space . The abstract notions of symplectic wave-packet groups and metaplectic wave-packet representations will be introduced. We then present an admissibility condition on closed subgroups of the real symplectic group , which guarantees the square-integrability of the associated metaplectic wave-packet representation on .

On polynomials with given Hilbert function and applications

Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur.

On stability of Leray's stationary solutions of the Navier–Stokes system in exterior domains

This paper studies the stability of a stationary solution of the Navier–Stokes system in $3$-D exterior domains. The stationary solution is called a Leray's stationary solution if the Dirichlet integral is finite. We apply an energy inequality and maximal $L^p$-in-time regularity for Hilbert space-valued functions to derive the decay rate with respect to time of energy solutions to a perturbed Navier–Stokes system governing a Leray's stationary solution.