Hilbert Function Research Articles

Distinguishing $\Bbbk$-configurations

A $\Bbbk$-configuration is a set of points $\mathbb{X}$ in $\mathbb{P}^{2}$ that satisfies a number of geometric conditions. Associated to a $\Bbbk$-configuration is a sequence $(d_{1},\ldots,d_{s})$ of positive integers, called its type, which encodes many of its homological invariants. We distinguish $\Bbbk$-configurations by counting the number of lines that contain $d_{s}$ points of $\mathbb{X}$. In particular, we show that for all integers $m\gg0$, the number of such lines is precisely the value of $\Delta\mathbf{H}_{m\mathbb{X}}(md_{s}-1)$. Here, $\Delta\mathbf{H}_{m\mathbb{X}}(-)$ is the first difference of the Hilbert function of the fat points of multiplicity $m$ supported on $\mathbb{X}$.

Effective Criteria for Specific Identifiability of Tensors and Forms

In applications where the tensor rank decomposition arises, one often relies on its identifiability properties for interpreting the individual rank-$1$ terms appearing in the decomposition. Several criteria for identifiability have been proposed in the literature, however few results exist on how frequently they are satisfied. We propose to call a criterion effective if it is satisfied on a dense, open subset of the smallest semi-algebraic set enclosing the set of rank-$r$ tensors. We analyze the effectiveness of Kruskal's criterion when it is combined with reshaping. It is proved that this criterion is effective for both real and complex tensors in its entire range of applicability, which is usually much smaller than the smallest typical rank. Our proof explains when reshaping-based algorithms for computing tensor rank decompositions may be expected to recover the decomposition. Specializing the analysis to symmetric tensors or forms reveals that the reshaped Kruskal criterion may even be effective up to the smallest typical rank for some third, fourth and sixth order symmetric tensors of small dimension as well as for binary forms of degree at least three. We extended this result to $4 \times 4 \times 4 \times 4$ symmetric tensors by analyzing the Hilbert function, resulting in a criterion for symmetric identifiability that is effective up to symmetric rank $8$, which is optimal.

Simplicial orders and chordality

Chordal clutters in the sense of Bigdeli et al. (J Comb Theory Ser A 145:129---149, 2017) and Morales et al. (Ann Fac Sci Toulouse Ser 6 23(4):877---891, 2014) are defined via simplicial orders. Their circuit ideal has a linear resolution, independent of the characteristic of the base field. We show that any Betti sequence of an ideal with linear resolution appears as the Betti sequence of the circuit ideal of such a chordal clutter. Associated with any simplicial order is a sequence of integers which we call the $$\lambda $$ź-sequence of the chordal clutter. All possible $$\lambda $$ź-sequences are characterized. They are intimately related to the Hilbert function of a suitable standard graded K-algebra attached to the chordal clutter. By the $$\lambda $$ź-sequence of a chordal clutter, we determine other numerical invariants of the circuit ideal, such as the $$\mathbf h $$h-vector and the Betti numbers.

Castelnuovo Mumford regularity with respect to multigraded ideals

In this article we extend a previous definition of Castelnuovo–Mumford regularity for modules over an algebra graded by a finitely generated abelian group.Our notion of regularity is based on Maclagan and Smith's definition, and is extended first by working over any commutative base ring, and second by considering local cohomology with support in an arbitrary finitely generated graded ideal B, obtaining, for each B, a B-regularity region. The first extension provides a natural approach for working with families of sheaves or of graded modules, while the second opens new applications.Even in the more restrictive framework where Castelnuovo–Mumford was defined before us, there were only very partial results on estimates for the shifts in a minimal graded free resolution from the Castelnuovo–Mumford regularity. We prove sharp estimates in our general framework, and this is one of our main advances.We provide tools to deduce information on the graded Betti numbers from the knowledge of regions where the local cohomology with support in a given graded ideal vanishes. Conversely, vanishing of local cohomology with support in any graded ideal is deduced from the shifts in a free resolution and the local cohomology of the polynomial ring. The flexibility of treating local cohomology with respect to any B opens up new possibilities for passing information.We provide new persistence results for the vanishing of local cohomology that extend the fact that weakly regular implies regular in the classical case, and we give sharp estimates for the regularity of a truncation of a module.In the last part, we present a result on Hilbert functions for multigraded polynomial rings, which provides a simple proof of the generalized Grothendieck–Serre formula.

Algebraic invariants of projective monomial curves associated to generalized arithmetic sequences

Let K be an infinite field and let m1<⋯<mn be a generalized arithmetic sequence of positive integers, i.e., there exist h,d,m1∈Z+ such that mi=hm1+(i−1)d for all i∈{2,…,n}. We consider the projective monomial curve C⊂PKn parametrically defined byx1=sm1tmn−m1,…,xn−1=smn−1tmn−mn−1,xn=smn,xn+1=tmn. In this work, we characterize the Cohen–Macaulay and Koszul properties of the homogeneous coordinate ring K[C] of C. Whenever K[C] is Cohen–Macaulay we also obtain a formula for its Cohen–Macaulay type. Moreover, when h divides d, we obtain a minimal Gröbner basis G of the vanishing ideal of C with respect to the degree reverse lexicographic order. From G we derive formulas for the Castelnuovo–Mumford regularity, the Hilbert series and the Hilbert function of K[C] in terms of the sequence.

The Corona Problem for Kernel Multiplier Algebras

We prove an alternate Toeplitz corona theorem for the algebras of pointwise kernel multipliers of Besov-Sobolev spaces on the unit ball in $\mathbb{C}^{n}$, and for the algebra of bounded analytic functions on certain strictly pseudoconvex domains and polydiscs in higher dimensions as well. This alternate Toeplitz corona theorem extends to more general Hilbert function spaces where it does not require the complete Pick property. Instead, the kernel functions $k_{x}\left(y\right)$ of certain Hilbert function spaces $\mathcal{H}$ are assumed to be invertible multipliers on $\mathcal{H}$, and then we continue a research thread begun by Agler and McCarthy in 1999, and continued by Amar in 2003, and most recently by Trent and Wick in 2009. In dimension $n=1$ we prove the corona theorem for the kernel multiplier algebras of Besov-Sobolev Banach spaces in the unit disk, extending the result for Hilbert spaces $H^\infty\cap Q_p$ by A. Nicolau and J. Xiao.

The regularity index of up to 2n − 1 equimultiple fat points of [formula omitted

Let X=mP1+⋯+mPn+k be a fat point subscheme of Pn, where Supp(X) consists of n+k distinct points which generate Pn. We study the regularity index τ(X) of X, which is the least degree in which the Hilbert function of X equals its Hilbert polynomial. We prove that the generalized Segre's bound for τ(X) holds if n≥4 and there are k+3 points of Supp(X) on a linear subspace Λ≃P3. We assume Supp(X) is not in general position and call d the least integer for which there exists a linear subspace Λ of dimension d containing at least d+2 points of Supp(X). We prove that the generalized Segre's bound holds for simple points when either 3≤k≤n+1 and d>k−3 or k=4 with no restriction on d. For m≥2 we prove the generalized Segre's bound when Supp(X) consists of n+4 points and either there are at least 3 points on a line or at least 5 points on a plane or at least 6 points on a linear subspace Λ≃P3. Finally we prove that, in general, 2m−1≤τ(X)≤2m when 3≤k≤n−1 and d>k−1, and we extend this result to the non-equimultiple case. We also provide cases in which the previous bound gives the generalized Segre's bound.

The EGH Conjecture and the Sperner property of complete intersections

Let A A be a graded complete intersection over a field and B B the monomial complete intersection with the generators of the same degrees as A A . The EGH conjecture says that if I I is a graded ideal in A A , then there should be an ideal J J in B B such that B / J B/J and A / I A/I have the same Hilbert function. We show that if the EGH conjecture is true, then it can be used to prove that every graded complete intersection over any field has the Sperner property.

Classifying local Artinian Gorenstein algebras

The classification of local Artinian Gorenstein algebras is equivalent to the study of orbits of a certain non-reductive group action on a polynomial ring. We give an explicit formula for the orbits and their tangent spaces. We apply our technique to analyse when an algebra is isomorphic to its associated graded algebra. We classify algebras with Hilbert function (1, 3, 3, 3, 1), obtaining finitely many isomorphism types, and those with Hilbert function (1, 2, 2, 2, 1, 1, 1). We consider fields of arbitrary, large enough, characteristic.

Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas

Abstract We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen–Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine their Hilbert functions and Castelnuovo–Mumford regularities. As a consequence, we find explicit minimal reductions for all Ferrers and many specialized Ferrers ideals, as well as their reduction numbers. These results can be viewed as extensions of the classical Dedekind–Mertens formula for the content of the product of two polynomials.

On Hadamard products of linear varieties

In this paper, we address the Hadamard product of linear varieties not necessarily in general position. In [Formula: see text], we obtain a complete description of the possible outcomes. In particular, in the case of two disjoint finite sets [Formula: see text] and [Formula: see text] of collinear points, we get conditions for [Formula: see text] to be either a collinear finite set of points or a grid of [Formula: see text] points. In [Formula: see text] under suitable conditions (which we prove to be generic), we show that [Formula: see text] consists of [Formula: see text] points on the two different rulings of a nondegenerate quadric and we compute its Hilbert function in the case [Formula: see text]

Extreme rays of Hankel spectrahedra for ternary forms

The cone of sums of squares is one of the central objects in convex algebraic geometry. Its defining linear inequalities correspond to the extreme rays of the dual convex cone. This dual cone is a spectrahedron, which can be explicitly realized as a section of the cone of positive semidefinite matrices with the linear subspace of Hankel (or middle catalecticant) matrices. In this paper we initiate a systematic study of the extreme rays of Hankel spectrahedra for ternary forms. We show that the Zariski closure of the union of extreme rays is the variety of all Hankel matrices of corank at least 4, an irreducible variety of codimension 10 and we determine its degree. We explicitly construct an extreme ray of maximal rank using the Cayley–Bacharach Theorem for plane curves. We apply our results to the study of the algebraic boundary of the cone of sums of squares. Its irreducible components are dual to varieties of Gorenstein ideals with certain Hilbert functions. We determine these Hilbert functions for some cases of small degree. We also observe surprising gaps in the ranks of Hankel matrices of the extreme rays.

Minimum distance functions of graded ideals and Reed–Muller-type codes

We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed–Muller-type codes over finite fields. This gives an algebraic formulation of the minimum distance of a projective Reed–Muller-type code in terms of the algebraic invariants and structure of the underlying vanishing ideal. Then we give a method, based on Gröbner bases and Hilbert functions, to find lower bounds for the minimum distance of certain Reed–Muller-type codes. Finally we show explicit upper bounds for the number of zeros of polynomials in a projective nested cartesian set and give some support to a conjecture of Carvalho, Lopez-Neumann and López.

Non-Normal Very Ample Polytopes – Constructions and Examples

ABSTRACTWe answer several questions posed by Beck, Cox, Delgado, Gubeladze, Haase, Hibi, Higashitani, and Maclagan in [Cox et al. 14, Question 3.5 (1),(2), Question 3.6], [Beck et al. 15, Conjecture 3.5(a),(b)], and [Hasse et al. 07, Open question 3 (a),(b) p. 2310, Question p. 2316] by constructing a new family of non-normal very ample polytopes. These polytopes are certain segmental fibrations of unimodular graph polytopes, we explicitly compute their invariants – Hilbert function, Ehrhart polynomial, and gap vector.

Hilbert functions of monomial ideals containing a regular sequence

Let M be an ideal in K[x1,...,x n] (K is a field) generated by products of linear forms and containing a homogeneous regular sequence of some length. We prove that ideals containing M satisfy the Eisenbud–Green–Harris conjecture and moreover prove that the Cohen–Macaulay property is preserved. We conclude that monomial ideals satisfy this conjecture. We obtain that the h-vector of Cohen–Macaulay simplicial complex Δ is the h-vector of Cohen–Macaulay (a 1 - 1,..., a t - 1)-balanced simplicial complex, where t is the height of the Stanley–Reisner ideal of Δ and (a 1,..., a t) is the type of some regular sequence contained in this ideal.

Residual intersections and the annihilator of Koszul homologies

Cohen-Macaulayness, unmixedness, the structure of the canonical module and the stability of the Hilbert function of algebraic residual intersections are studied in this paper. Some conjectures about these properties are established for large classes of residual intersections without restricting local number of generators of the ideals involved. A family of approximation complexes for residual intersections is constructed to determine the above properties. Moreover some general properties of the symmetric powers of quotient ideals are determined which were not known even for special ideals with a small number of generators. Acyclicity of a prime case of these complexes is shown to be equivalent to find a common annihilator for higher Koszul homologies. So that, a tight relation between residual intersections and the uniform annihilator of positive Koszul homologies is unveiled that sheds some light on their structure.

Volume and Hilbert functions of R-divisors

Let X be a proper, normal algebraic variety of dimension n over a field K and D an R-divisor on X. The Hilbert function of D is the function H(X,D) : m 7→ h(mD) := dimK H(X,OX(bmDc)); defined for all m ∈ R. If D is an ample Cartier divisor then H(X,D) agrees with the usual Hilbert polynomial whenever m 1 is an integer, but in general H(X,D) is not a polynomial, not even if D is a Z-divisor and m ∈ Z. The simplest numerical invariant associated to the Hilbert function is the volume of D, defined as

ON HILBERT FUNCTIONS OF GENERAL INTERSECTIONS OF IDEALS

Let $I$ and $J$ be homogeneous ideals in a standard graded polynomial ring. We study upper bounds of the Hilbert function of the intersection of $I$ and $g(J)$, where $g$ is a general change of coordinates. Our main result gives a generalization of Green’s hyperplane section theorem.

Star-Configurations in ℙn and the Weak-Lefschetz Property

We find that some union of two star-configurations in ℙ2 has generic Hilbert function. Applying the result, we prove that some Artinian quotients of a coordinate ring of a star-configuration in ℙn satisfy the weak-Lefschetz property. More precisely, let 𝕏 and 𝕐 be star-configurations in ℙ2 of type (2, s) and (2, s + 1) defined by forms F1,…, Fs, and G1,…, Gs, L, respectively, with deg(Fi) = deg(Gi) ≤2 for i = 1,…, s and s ≥ 3. If L is a general linear form in R = 𝕜[x0, x1, x2], then R/(I𝕏 + I𝕐) has the weak-Lefschetz property with a Lefschetz element L, which extends the result of [21].

Hilbert polynomials of j-transforms

AbstractWe study transformations of finite modules over Noetherian local rings that attach to a module M a graded module H(x)(M) defined via partial systems of parameters x of M. Despite the generality of the process, which are called j-transforms, in numerous cases they have interesting cohomological properties. We focus on deriving the Hilbert functions of j-transforms and studying the significance of the vanishing of some of its coefficients.