Hilbert Function Research Articles

On the Analytic Type of Artin Algebras

In this paper we give a complete analytic classification of Artin Gorenstein almost stretched algebras, i.e., Artin Gorenstein algebras with Hilbert function {1, n, 2,…, 2, 1,…, 1}.

Multigraded Hilbert functions and toric complete intersection codes

Let X be a complete n-dimensional simplicial toric variety with homogeneous coordinate ring S. We study the multigraded Hilbert function HY of reduced 0-dimensional subschemes Y in X. We provide explicit formulas and prove non-decreasing and stabilization properties of HY when Y is a 0-dimensional complete intersection in X. We apply our results to computing the dimension of some evaluation codes on 0-dimensional complete intersection in simplicial toric varieties.

Sums of squares on the hypercube

Let X be a finite set of points in \({\mathbb {R}}^n\). A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we are interested in rational function representations of small degree. We derive a general upper bound in terms of the Hilbert function of X, and we show that this upper bound is tight for the case of quadratic functions on the hypercube \(C=\{0,1\}^n\), a very well studied case in combinatorial optimization. Using the lower bounds for C we construct a family of globally nonnegative quartic polynomials, which are not sums of squares of rational functions of small degree. To our knowledge this is the first construction for Hilbert’s 17th problem of a family of polynomials of bounded degree which need increasing degrees in rational function representations as the number of variables n goes to infinity. We note that representation theory of the symmetric group \(S_n\) plays a crucial role in our proofs of the lower bounds.

Hilbert–Samuel functions of well bifiltered modules

In an earlier paper, we studied the Hilbert quasi-polynomial functions of finitely generated bigraded modules in the general framework when the base ring is bigraded and generated by finitely many homogeneous elements of arbitrary degrees. In this paper, we introduce the concept of [Formula: see text]-good bifiltration [Formula: see text] on a finitely generated [Formula: see text]-module [Formula: see text], where [Formula: see text] and [Formula: see text] are specified noetherian filtrations on the noetherian ring [Formula: see text]. The bigraded modules associated with such bifiltrations are shown to be finitely generated under reasonable hypotheses. Their Hilbert functions are studied. The Hilbert–Samuel function of [Formula: see text] with respect to the [Formula: see text]-good bifiltration [Formula: see text] of [Formula: see text] is one of them. It is proved, among others, that this function is a quasi-polynomial function in two variables and that if [Formula: see text] is a noetherian local ring and if the filtrations [Formula: see text] and [Formula: see text] are primary filtrations, then its degree equals the Krull dimension of [Formula: see text].

Hilbert functions of [formula omitted]-stable artinian Gorenstein algebras

We describe the graded characters and Hilbert functions of certain graded artinian Gorenstein quotients of the polynomial ring which are also representations of the symmetric group. Specifically, we look at those algebras whose socles are trivial representations and whose principal apolar submodules are generated by the sum of the orbit of a power of a linear form.

On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence

Let $A$ be a local Artinian Gorenstein algebra with maximal ideal $\fM $, \[P_A(z) := \sum _{p=0}^{\infty } (\tor _p^A(k,k))z^p \] its Poicar\'{e} series. We prove that $P_A(z)$ is rational if either $\dim _k({\fM ^2/\fM ^3}) \leq 4 $ and $ \dim _k(A) \leq 16,$ or there exist $m\leq 4$ and $c$ such that the Hilbert function $H_A(n)$ of $A$ is equal to $ m$ for $n\in [2,c]$ and equal to $1$ for $n =c+1$. The results are obtained due to a decomposition of the apolar ideal $\Ann (F)$ when $F=G+H$ and $G$ and $H$ belong to polynomial rings in different variables.

On a differential game in a system with distributed parameters

We consider a game problem of approach for a system whose dynamics is described by a partial differential equation not of Kovalevskaya type, i.e., unsolved with respect to the time derivative. The equation with boundary conditions is written in a Hilbert function space in an abstract form as a differential operator equation. Using the method of resolving functionals, we obtain sufficient conditions for the approach of the system’s dynamical vector to a cylindrical terminal set. The results are exemplified by means of a model problem concerning a filtering process for a fluid in fractured porous rocks.

Extensors and the Hilbert scheme

The Hilbert scheme $Hilb_{p(t)}^n$ parametrizes closed subschemes and families of closed subschemes in the projective space $P^n$ with a fixed Hilbert polynomial $p(t)$. It is classically realized as a closed subscheme of a Grassmannian or a product of Grassmannians. In this paper we present a method that allows to derive scheme theoretical global equations for $Hilb_{p(t)}^n$ in the Plucker coordinates of a Grassmannian $Grass_p^N$, where $p$ and $N$ depend on the dimension $n$ of the projective space and on the Hilbert polynomial $p(t)$. Using this method we obtain the already known set of equations given by Iarrobino and Kleiman in 1999, the one conjectured by Bayer in 1982 and proved by Haiman and Sturmfels in 2004, and also a new set of equations of degree lower than the previous ones. The novelties of our approach are essentially two. The first one is a local study of the Hilbert functor through special sets of open subfunctors obtained exploiting the symmetries of the Hilbert scheme and the combinatorial properties of monomial ideals, mainly the Borel-fixed ones. The second one is a generalization of the theory of extensors to the setting of free modules over any ring $A$ and the description of any exterior product of elements of a free submodule in terms of Plucker coordinates.

Double-generic initial ideal and Hilbert scheme

Following the approach in the book "Commutative Algebra", by D. Eisenbud, where the author describes the generic initial ideal by means of a suitable total order on the terms of an exterior power, we introduce first the generic initial extensor of a subset of a Grassmannian and then the double-generic initial ideal of a so-called GL-stable subset of a Hilbert scheme. We discuss the features of these new notions and introduce also a partial order which gives another useful description of them. The double-generic initial ideals turn out to be the appropriate points to understand some geometric properties of a Hilbert scheme: they provide a necessary condition for a Borel ideal to correspond to a point of a given irreducible component, lower bounds for the number of irreducible components in a Hilbert scheme and the maximal Hilbert function in every irreducible component. Moreover, we prove that every isolated component having a smooth double-generic initial ideal is rational. As a byproduct, we prove that the Cohen-Macaulay locus of the Hilbert scheme parameterizing subschemes of codimension 2 is the union of open subsets isomorphic to affine spaces. This improves results by J. Fogarty (1968) and R. Treger (1989).

On the Hilbert function of lines union one non-reduced point

Polito SFX(opens in a new window)| Export | Download | Add to List | More... Annali della Scuola normale superiore di Pisa - Classe di scienze Volume 15, 2016, Pages 69-84 On the Hilbert function of lines union one non-reduced point (Article) Carlini, E.a , Catalisano, M.V.bc , Geramita, A.V.d a Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca Degli Abbruzzi 24, Torino, Italy b DIME Dipartimento di Ingegneria Meccanica Energetica Gestionale E Dei Trasporti, Universita Degli Studi di Genova, Piazzale Kennedy pad. D, Genova, Italy c Department of Mathematics and Statistics, Queen's University, Kingston, ON, Canada View additional affiliations View references (18) Abstract In this paper we consider the problem of determining the Hilbert function of schemes XcP which are the generic union of s lines and one m- multiple point. We completely solve this problem for any s and m when n ≥ 4. When n = 3 we find several defective such schemes and conjecture that they are the only ones. We verify this conjecture in several cases.

Real-dihedral harmonic Maass forms and CM-values of Hilbert modular functions

In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [CM-values of Hilbert modular functions, Invent. Math.163(2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

Exact pairs of homogeneous zero divisors

Let S be a standard graded Artinian algebra over a field k. We identify constraints on the Hilbert function of S which are imposed by the hypothesis that S contains an exact pair of homogeneous zero divisors. As a consequence, we prove that if S is a compressed level algebra, then S does not contain any homogeneous zero divisors.

On the normalized arithmetic Hilbert function

Let $X$ be a subvariety of dimension n of the projective space over $\overline{\mathbb{Q}}$, and $H_{norm}(X;D)$ the normalized arithmetic Hilbert function of $X$ introduced by Philippon and Sombra. We show that this function admits the following asymptotic expansion $H_{norm}(X;D) = \frac{\widehat{h}(X)}{(n + 1)!}D^{n+1} + o(D^{n+1})$ where $\widehat{h}(X)$ is the normalized height of $X$. This gives a positive answer to a question raised by Philippon and Sombra.

Mini-Workshop: Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems

It is a fundamental challenge for many problems of significant current interest in algebraic geometry and commutative algebra to understand symbolic powers I^{(m)} of homogeneous ideals I in polynomial rings, particularly ideals of linear varieties. Such problems include computing Waring ranks of polynomials, determining the occurrence of equality I^{(m)} = I^m (or, more generally, of containments I^{(m)} \subseteq I^r ), computing Waldschmidt constants (i.e., determining the limit of the ratios of the least degree of an element in I^{(m)} to the least degree of an element of I^m ), and studying major conjectures such as Nagata’s Conjecture and the uniform SHGH Conjecture (which respectively specify the Waldschmidt constant of ideals of generic points in the plane and the Hilbert functions of their symbolic powers).

The unimodality of pure $O$-sequences of type two in four variables

Since the 1970’s, great interest has been taken in the study of pure O-sequences, which, due to Macaulay’s theory of inverse systems, have a bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these according to their shape. Macaulay’s theorem immediately gives us that all Artinian algebras in two variables have unimodal Hilbert functions. Furthermore, it has been shown that all Artinian level monomial algebras of type two in three variables have unimodal Hilbert functions. This paper will classify all Artinian level monomial algebras of type two in four variables into two classes of ideals, prove that they are strictly unimodal and show that one of the classes is licci.

On the Hilbert quasi-polynomials for non-standard graded rings

The Hilbert function, its generating function and the Hilbert polynomial of a graded ring K[ x 1 , ..., x k ] have been extensively studied since the famous paper of Hilbert: Ueber die Theorie der algebraischen Formen ( [Hilbert, 1890]) . In particular, the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry. If the ring grading is non-standard, then its Hilbert function is not eventually equal to a polynomial but to a quasi-polynomial. It turns out that a Hilbert quasi-polynomial P of degree n splits into a polynomial S of degree n and a lower degree quasi-polynomial T . We have completely determined the degree of T and the first few coefficients of P . Moreover, the quasi-polynomial T has a periodic structure that we have described. We have also developed a software to compute effectively the Hilbert quasi-polynomial for any ring K[ x 1 , ..., x k ]/ I .

On the Hilbert function of the tangent cone of a monomial curve

In this paper we study the Hilbert function of \(\mathrm {gr}_{\mathfrak {m}}(R)\), when R is a numerical semigroup ring or, equivalently, the coordinate ring of a monomial curve. In particular, we prove a sufficient condition for a numerical semigroup ring in order get a non-decreasing Hilbert function, without making any assumption on its embedding dimension; moreover, we show how this new condition allows us to improve known results about this problem. To this end we use certain invariants of the semigroup, with particular regard to its Apery-set.

Hilbert polynomials of multigraded filtrations of ideals

Hilbert functions and Hilbert polynomials of Zs-graded admissible filtrations of ideals {F(n_)}n_∈Zs such that λ(RF(n_)) is finite for all n_∈Zs are studied. Conditions are provided for the Hilbert function HF(n_):=λ(R/F(n_)) and the corresponding Hilbert polynomial PF(n_) to be equal for all n_∈Ns. A formula for the difference HF(n_)−PF(n_) in terms of local cohomology of the extended Rees algebra of F is proved which is used to obtain sufficient linear relations analogous to the ones given by Huneke and Ooishi among coefficients of PF(n_) so that HF(n_)=PF(n_) for all n_∈Ns. A theorem of Rees about joint reductions of the filtration {IrJs‾}r,s∈Z is generalised for admissible filtrations of ideals in two-dimensional Cohen–Macaulay local rings. Necessary and sufficient conditions are provided for the multi-Rees algebra of an admissible Z2-graded filtration F to be Cohen–Macaulay.

Proof of Stembridge’s conjecture on stability of Kronecker coefficients

We prove a conjecture of Stembridge concerning stability of Kronecker coefficients that vastly generalizes Murnaghan’s theorem. The main idea is to identify the sequences of Kronecker coefficients in question with Hilbert functions of modules over finitely generated algebras. The proof only uses Schur–Weyl duality and the Borel–Weil theorem and does not rely on any existing work on Kronecker coefficients.

Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space

In this paper, approximate solutions to a class of fractional differential equations with delay are presented by using a semi-analytical approach in Hilbert function space. Further, the uniqueness of the solution is proved in the space of real-valued continuous functions, as well as the existence of the solution is proved in Hilbert function space. We also prove convergence and perform an analysis error for the proposed approach. Sophisticated delay differential equations of fractional order are considered as test examples. Numerical results illustrate the efficiency of the proposed approach computationally.