Hilbert Function Research Articles

Signature-based algorithm under non-compatible term orders and its application to change of ordering

The notion of the compatibility between a term order in a polynomial ring R and a module term order in Rl is crucial to ensure the termination of a signature-based algorithm for general input ideals. However, it is shown experimentally that the compatibility does not necessarily imply efficient computation. Our experiments show that combining non-compatible term orders can improve performance for computing Gröbner bases with respect to some term orders. In such cases, we can use the Hilbert function to guarantee the termination. The Hilbert function can be computed by using a Gröbner basis with respect to some term order and thus the resulting algorithm is considered a change of ordering algorithm. In this paper, we give the details of the new change of ordering algorithm and we compare its performance with that of the usual Hilbert-driven Buchberger algorithm and the Gröbner walk algorithm.

Differential theory of zero-dimensional schemes

To study a 0-dimensional scheme X in Pn over a perfect field K, we use the module of Kähler differentials ΩR/K1 of its homogeneous coordinate ring R and its exterior powers, the higher modules of Kähler differentials ΩR/Km. One of our main results is a characterization of weakly curvilinear schemes X by the Hilbert polynomials of the modules ΩR/Km which allows us to check this property algorithmically without computing the primary decomposition of the vanishing ideal of X. Further main achievements are precise formulas for the Hilbert functions and Hilbert polynomials of the modules ΩR/Km for a fat point scheme X which extend and settle previous partial results and conjectures. Underlying these results is a novel method: we first embed the homogeneous coordinate ring R into its truncated integral closure R˜. Then we use the corresponding map from the module of Kähler differentials ΩR/K1 to ΩR˜/K1 to find a formula for the Hilbert polynomial HP(ΩR/K1) and a sharp bound for the regularity index ri(ΩR/K1). Next we extend this to formulas for the Hilbert polynomials HP(ΩR/Km) and bounds for the regularity indices of the higher modules of Kähler differentials. As a further application, we characterize uniformity conditions on X using the Hilbert functions of the Kähler differential modules of X and its subschemes.

Products and powers of principal symmetric ideals

Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.

On the monotonicity of the Hilbert functions for 4-generated pseudo-symmetric monomial curves

In this article we solve the following problem: “The Hilbert function of the local ring of a 4-generated pseudo-symmetric numerical semigroup is always non-decreasing.” We give a complete characterization of the standard bases when the tangent cone is not Cohen-Macaulay by showing that the number of elements in the standard basis depends on some parameters we define. Since the tangent cone is not Cohen-Macaulay, non-decreasingness of the Hilbert function was not guaranteed, thus we proved the non-decreasingness from our explicit Hilbert function computation.

CB-HDM: ECG signal based heart disease classification using convolutional block attention assisted hybrid deep Maxout network

Cardiovascular diseases (CVD) stand as the leading cause of death globally. Diagnosis of CVD relies on electrocardiograms (ECGs), which record the heart’s activity. Adequate labeled data is essential for effectively training machine learning techniques, which may also demand considerable time and memory resources. To address these existing issues, this research introduces an improved detection technique for predicting CVD. Initially, data is sourced from the MIT-BIH arrhythmia database, followed by pre-processing using the Modified FIR filter model (M-FIR) to filter ECG signals. It is used to enhance signal quality and eliminate unwanted noise. M-FIR is used to preserve the specific features of the signal. Feature extraction utilizes the Radial Hilbert Function Transform Network (RHFTN) technique. It can extract the temporal and spectral signal features for effective CVD detection. The Improved Bald Eagle Search Algorithm (IBES) is employed to select features and minimize irrelevant features. Detection and classification of heart disease from ECG signals are achieved using the Convolutional Block Attention Assisted Hybrid Deep Maxout Network model (CB-HDM). Losses in the network model are mitigated by the Gazelle Optimization Algorithm (GOA). The evaluation results are simulated using Python. The proposed model achieved 98.81% accuracy, 96.18% precision, 96.87% recall, and a 94.39% F1 score. The proposed model demonstrates superior efficiency compared to other classifiers.

The Variety of Polar Simplices II

We discuss the space VPS(Q, H) of ideals with Hilbert function H = ( 1 , n , n , … ) that are apolar to a full rank quadric Q. We prove that its components of saturated ideals are closely related to the locus of Gorenstein algebras and to the Slip component in border apolarity. We also point out an important error in [20] and provide the necessary corrections.

Lattice points in slices of prisms

Abstract We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov; moreover, they coincide with polymatroids satisfying the strong exchange property up to an affinity. We give a combinatorial formula for all the Ehrhart coefficients in terms of the number of weighted permutations satisfying certain compatibility properties. This result proves that all these polytopes are Ehrhart positive. Additionally, via an extension of a result by Early and Kim, we give a combinatorial interpretation for all the coefficients of the $h^*$ -polynomial. All of our results provide a combinatorial understanding of the Hilbert functions and the h-vectors of all algebras of Veronese type, a problem that had remained elusive up to this point. A variety of applications are discussed, including expressions for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; some extensions of Laplace’s result on the combinatorial interpretation of the volume of the hypersimplex; a multivariate generalization of the flag Eulerian numbers and refinements; and a short proof of the Ehrhart positivity of the independence polytope of all uniform matroids.

Multigraded algebras and multigraded linear series

AbstractThis paper is devoted to the study of multigraded algebras and multigraded linear series. For an ‐graded algebra , we define and study its volume function , which computes the asymptotics of the Hilbert function of . We relate the volume function to the volume of the fibers of the global Newton–Okounkov body of . Unlike the classical case of standard multigraded algebras, the volume function is not a polynomial in general. However, in the case when the algebra has a decomposable grading, we show that the volume function is a polynomial with nonnegative coefficients. We then define mixed multiplicities in this case and provide a full characterization for their positivity. Furthermore, we apply our results on multigraded algebras to multigraded linear series. Our work recovers and unifies recent developments on mixed multiplicities. In particular, we recover results on the existence of mixed multiplicities for (not necessarily Noetherian) graded families of ideals and on the positivity of the multidegrees of multiprojective varieties.

The Hilbert function of general unions of lines and double lines in the projective space

We study the Hilbert function of a general union X⊂P3 of x double lines and y lines. In many cases (e.g. always for x=2 and y≥3 or for x=3 and y≥2 or for x≥4 and y≥⌈((3x+43)−27x−12)/(3x+2)⌉+3−x) we prove that X has maximal rank. We give a few examples of x and y for which X has not maximal rank.

Chern character, infinitesimal Abel-Jacobi map and semi-regularity map

Using Chern character, we construct a natural transformation from the local Hilbert functor to a functor of Artin rings defined from Hochschild homology. This enables us to realize (after slight modification) the infinitesimal Abel-Jacobi map as a morphism between tangent spaces of two functors of Artin rings and also enables us to reconstruct the semi-regularity map together with giving a different proof of a theorem of Bloch stating that the semi-regularity map annihilates certain obstructions to embedded deformations of a closed subvariety which is a locally complete intersection.

Perazzo hypersurfaces and the weak Lefschetz property

We deal with Perazzo hypersurfaces X=V(f) in Pn+2 defined by a homogeneous polynomial f(x0,x1,…,xn,u,v)=p0(u,v)x0+p1(u,v)x1+⋯+pn(u,v)xn+g(u,v), where p0,p1,…,pn are algebraically dependent but linearly independent forms of degree d−1 in K[u,v] and g is a form in K[u,v] of degree d. Perazzo hypersurfaces have vanishing hessian and, hence, the associated graded artinian Gorenstein algebra Af fails the strong Lefschetz property. In this paper, we first determine the maximum and minimum Hilbert function of Af, we prove that the Hilbert function of Af is always unimodal and we determine when Af satisfies the weak Lefschetz property. We illustrate our results with many examples and we show that our results do not generalize to Perazzo hypersurfaces X=V(f) in Pn+3 defined by a homogeneous polynomial f(x0,x1,…,xn,u,v,w)=p0(u,v,w)x0+p1(u,v,w)x1+⋯+pn(u,v,w)xn+g(u,v,w), where p0,p1,…,pn are algebraically dependent but linearly independent forms of degree d−1 in K[u,v,w] and g is a form in K[u,v,w] of degree d.

On the Hilbert Function of General Unions of Curves in Projective Spaces

On the Hilbert Function of General Unions of Curves in Projective Spaces

Unique maximal Betti diagrams for Artinian Gorenstein k-algebras with the weak Lefschetz property

We give an alternate proof for a theorem of Migliore and Nagel. In particular, we show that if H is an SI-sequence, then the collection of Betti diagrams for all Artinian Gorenstein k-algebras with the weak Lefschetz property and Hilbert function H has a unique largest element.

On the Stability of Multigraded Betti Numbers and Hilbert Functions

On the Stability of Multigraded Betti Numbers and Hilbert Functions

Limits of graded Gorenstein algebras of Hilbert function $$(1,3^k,1)$$

Limits of graded Gorenstein algebras of Hilbert function $$(1,3^k,1)$$

On minimal Gorenstein Hilbert functions

We conjecture that a class of Artinian Gorenstein Hilbert algebras called full Perazzo algebras always have minimal Hilbert function, fixing codimension and length. We prove the conjecture in length four and five, in low codimension. We also prove the conjecture for a particular subclass of algebras that occurs in every length and certain codimensions. As a consequence of our methods we give a new proof of part of a known result about the asymptotic behavior of the minimum entry of a Gorenstein Hilbert function.

On the Hilbert function of a finite scheme contained in a quadric surface

On the Hilbert function of a finite scheme contained in a quadric surface

Integrally Closed Ideals of Reduction Number Three

In a Cohen-Macaulay local ring [Formula: see text], we study the Hilbert function of an integrally closed [Formula: see text]-primary ideal [Formula: see text] whose reduction number is three. With a mild assumption we give an inequalityi [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th Hilbert coefficient and [Formula: see text] denotes a minimal reduction of [Formula: see text]. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore, this inequality becomes an equality if and only if the depth of the associated graded ring of [Formula: see text] is larger than or equal to [Formula: see text]. We also study the Cohen-Macaulayness of the associated graded rings of determinantal rings.

Number of generators of ideals in Jordan cells of the family of graded Artinian algebras of height two

We let A=R/I be a standard graded Artinian algebra quotient of R=k[x,y], the polynomial ring in two variables over a field k by an ideal I, and let n be its vector space dimension. The Jordan type Pℓ of a linear form ℓ∈A1 is the partition of n determining the Jordan block decomposition of the multiplication on A by ℓ – which is nilpotent. The first three authors previously determined which partitions of n=dimk⁡A may occur as the Jordan type for some linear form ℓ on a graded complete intersection Artinian quotient A=R/(f,g) of R, and they counted the number of such partitions for each complete intersection Hilbert function T[1].We here consider the family GT of graded Artinian quotients A=R/I of R=k[x,y], having arbitrary Hilbert function H(A)=T. The Jordan cell V(EP) corresponding to a partition P having diagonal lengths T is comprised of all ideals I in R whose initial ideal is the monomial ideal EP determined by P. These cells give a decomposition of the variety GT into affine spaces. We determine the generic number κ(P) of generators for the ideals in each cell V(EP), generalizing a result of [1]. In particular, we determine those partitions for which κ(P)=κ(T), the generic number of generators for an ideal defining an algebra A in GT. We also count the number of partitions P of diagonal lengths T having a given κ(P). A main tool is a combinatorial and geometric result allowing us to split T and any partition P of diagonal lengths T into simpler Ti and partitions Pi, such that V(EP) is the product of the cells V(EPi), and Ti is single-block: GTi is a Grassmannian.

The Hilbert function of general unions of lines, double lines and double points in projective spaces

The Hilbert function of general unions of lines, double lines and double points in projective spaces