Hilbert Basis Research Articles

Hilbert basis of the Lipman semigroup

In this Note, we give a new method to compute the Hilbert basis of the semigroup of certain positive divisors supported on the exceptional divisor of a normal surface singularity. Our approach is purely combinatorial and enables us to avoid the long calculation of the invariants of the ring as it is presented in the work of Altınok and Tosun.

Algebraic combinatorics of diametric magic circles

Abstract We provide a definition of a diametric magic circle of order n. In this paper, we use techniques in computational algebraic combinatorics and enumerative geometry to construct and to count d-magic circles. We also provide a description of its minimal Hilbert basis and determine the symmetry operations on d-magic circles. Finally, we give an algorithm for writing a natural d-magic circle in terms of the minimal Hilbert basis.

Valence-band of cubic semiconductors: Clifford algebra approach II

Application of Clifford algebra in the analysis of valence-band spin properties in semiconductors is considered. In the first part (Dargys A 2009 Phys. Scr.80 065701), for this purpose the isomorphism between multivectors and their matrix representations was used to transform the problem to Clifford algebra. Here equivalence rules are established between the spinors of Hilbert space and basis elements of the five-dimensional Clifford algebra Cl4, 1. Then, the rules are applied to the total angular momentum components and the two-band hole Hamiltonian. The resulting biquaternionic Schrödinger equation for hole spin is solved as an example.

Alternatives for testing total dual integrality

In this paper we provide characterizing properties of totally dual integral (TDI) systems, among others the following: a system of linear inequalities is TDI if and only if its coefficient vectors form a Hilbert basis, and there exists a test-set for the system’s dual integer programs where all test vectors have positive entries equal to 1. Reformulations of this provide relations between computational algebra and integer programming and they contain Applegate, Cook and McCormick’s sufficient condition for the TDI property and Sturmfels’ theorem relating toric initial ideals generated by square-free monomials to unimodular triangulations. We also study the theoretical and practical efficiency and limits of the characterizations of the TDI property presented here. In the particular case of set packing polyhedra our results correspond to endowing the weak perfect graph theorem with an additional, computationally interesting, geometric feature: the normal fan of the stable set polytope of a perfect graph can be refined into a regular triangulation consisting only of unimodular cones.

Ehrhart Clutters: Regularity and Max-Flow Min-Cut

If $\mathcal{C}$ is a clutter with $n$ vertices and $q$ edges whose clutter matrix has column vectors ${\mathcal A} = \{v_1, \ldots, v_q\}$, we call $\mathcal{C}$ an Ehrhart clutter if $\{(v_1,1),\ldots,(v_q,1)\} \subset \{ 0,1 \}^{n+1}$ is a Hilbert basis. Letting $A(P)$ be the Ehrhart ring of $P={\rm conv}(\mathcal{A})$, we are able to show that if $\mathcal{C}$ is a uniform unmixed MFMC clutter, then $\mathcal{C}$ is an Ehrhart clutter and in this case we provide sharp upper bounds on the Castelnuovo-Mumford regularity and the $a$-invariant of $A(P)$. Motivated by the Conforti-Cornuéjols conjecture on packing problems, we conjecture that if $\mathcal{C}$ is both ideal and the clique clutter of a perfect graph, then $\mathcal{C}$ has the MFMC property. We prove this conjecture for Meyniel graphs by showing that the clique clutters of Meyniel graphs are Ehrhart clutters. In much the same spirit, we provide a simple proof of our conjecture when $\mathcal{C}$ is a uniform clique clutter of a perfect graph. We close with a generalization of Ehrhart clutters as it relates to total dual integrality.

Normaliz: Algorithms for affine monoids and rational cones

Normaliz is a program for the computation of Hilbert bases of rational cones and the normalizations of affine monoids. It may also be used for solving diophantine linear systems. In this paper we present the algorithms implemented in the program.

A Macaulay2 interface for Normaliz

Normaliz is a tool for the computation of Hilbert bases of normal affine monoids and related tasks. We describe the Macaulay2 interface to Normaliz. It makes Normaliz accessible for operations on monoid algebras in Macaulay2 and, conversely, makes Macaulay2 a front-end for

Some inequalities involving a fractal operator of functions on the sphere

This paper undertakes the construction of fractal versions of the classical spherical harmonics. Some inequalities satisfied by the coefficients of the iterated function systems defin- ing the fractal functions provide sufficient conditions for the existence of new Hilbert bases of functions on the sphere. This fact confirms their properties of good approximation. The method- ology used implies the definition of an operator mapping standard functions into their fractal analogues. The transformation is linear and bounded and some upper bounds of its norm are also established.

Recognizing conic TDI systems is hard

In this note we prove that the problem of deciding whether or not a set of integer vectors forms a Hilbert basis is co-NP-complete. Equivalently, deciding whether a conic linear system is totally dual integral or not, is co-NP-complete. These statements are true even if the vectors in the set or respectively the coefficient vectors of the inequalities are 0–1 vectors having at most three ones.

Algebraic and combinatorial properties of ideals and algebras of uniform clutters of TDI systems

Let $\mathcal{C}$ be a uniform clutter and let A be the incidence matrix of $\mathcal{C}$ . We denote the column vectors of A by v 1,…,v q . Under certain conditions we prove that $\mathcal{C}$ is vertex critical. If $\mathcal{C}$ satisfies the max-flow min-cut property, we prove that A diagonalizes over ℤ to an identity matrix and that v 1,…,v q form a Hilbert basis. We also prove that if $\mathcal{C}$ has a perfect matching such that $\mathcal{C}$ has the packing property and its vertex covering number is equal to 2, then A diagonalizes over ℤ to an identity matrix. If A is a balanced matrix we prove that any regular triangulation of the cone generated by v 1,…,v q is unimodular. Some examples are presented to show that our results only hold for uniform clutters. These results are closely related to certain algebraic properties, such as the normality or torsion-freeness, of blowup algebras of edge ideals and to finitely generated abelian groups. They are also related to the theory of Gröbner bases of toric ideals and to Ehrhart rings.

Scheduling Partial Round Robin Tournaments Subject to Home Away Pattern Sets

We consider the following sports scheduling problem. Consider $2n$ teams in a sport league. Each pair of teams must play exactly one match in $2n-1$ days. That is, $n$ games are held simultaneously in a day. We want to make a schedule which has $n(2n-1)$ games for $2n-1$ days. When we make a schedule, the schedule must satisfy a constraint according to the HAP set, which designates a home game or an away game for each team and each date. Two teams cannot play against each other unless one team is assigned to a home game and the other team is assigned to an away game. Recently, D. Briskorn proposed a necessary condition for an HAP set to have a proper schedule. And he proposed a conjecture that such a condition is also sufficient. That is, if a solution to the linear inequalities exists, they must have an integral solution. In this paper, we rewrite his conjecture by using perfect matchings. We consider a monoid in the affine space generated by perfect matchings. In terms of the Hilbert basis of such a monoid, the problem is naturally generalized to a scheduling problem for not all pairs of teams described by a regular graph. In this paper, we show a regular graph such that the corresponding linear inequalities have a solution but do not have any integral solution. Moreover we discuss for which regular graphs the statement generalizing the conjecture holds.

Polyhedra: A package for computations with convex polyhedral objects

Convex polyhedral objects play an important role in numerous areas of algebraic geometry. The package Polyhedra enables one to work with such convex polyhedral objects in Macaulay 2. In particular, Polyhedra implements objects representing polyhedra, cones, and fans. Many standard operations, such as intersections, duality, convex hulls, and calculation of Hilbert bases are included. The package also provides functions to create convex polyhedral objects from some algebraic objects. THE PACKAGE AND ITS APPLICATIONS. Many objects in algebraic geometry can be described by convex polyhedral objects. For example, the Grobner degenerations of a homogeneous ideal are in one-to-one correspondence with the faces of its state polytope; see (St, Chapter 2). Geometric invariant theory (GIT) quotients coming from linearizations of the trivial bundle provide a second class of examples. For an effective action of the torus (C ) k on affine space A n , there is an associated exact sequence of lattices 0! K! Z n ! Z k ! 0. The GIT-quotients of this action are given by the normal fans of certain polyhedra in the associated Q-vector space KQ; for more information see (B, §III.2). The package Polyhedra provides methods within Macaulay 2 (M2) for creating and manipulating convex polyhedral objects such as polyhedra and cones. Some objects from algebraic geometry can also be translated into their corresponding convex polyhedral objects.

Polytopes of Magic Labelings of Graphs and the Faces of the Birkhoff Polytope

In this article, we construct and enumerate magic labelings of graphs using Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes. This enables us to generate and enumerate perfect matchings of a graph via magic labelings of the graph. We explore the correspondence of magic labelings of graphs with magic squares and define polytopes of magic labelings to give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.

A matroid-friendly basis for the quasisymmetric functions

A new Z -basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the quasisymmetric functions associated to matroids by the Hopf algebra morphism F of Billera, Jia, and Reiner [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv: math.CO/0606646]. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F distinguishes isomorphism classes of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer to the Hilbert basis question raised in [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv: math.CO/0606646] is given.

Hilbert Bases for Orthogonal Arrays

In this paper, we relate the problem of generating all 2-level orthogonal arrays of given dimension and strength, i.e. elements in OA (N,2n,m), where N is the number of rows, n is the number of factors, and m the strength, to the solution of an integer programming problem involving rational convex cones. We admit any number of replications in the arrays, i.e. we consider not only set but also multi-set. This problem can be theoretically solved by means of Hilbert bases which form a finite generating set for all the elements in the infinite set OA (N,2n,m), N ∈ ℕ. We discuss some examples which are explicitly solved with a software performing Hilbert bases computation.

Feasible region approximation: a comparison of search cone and convex hull methods

The concept of the design space encapsulates the search for optimal design configurations of engineering systems. During this search, infeasible or unsafe regions may be encountered, representing particular combinations of parameter values which cause the system to fail. These regions need to be characterized so that they can be avoided during the search for improved designs. In addition, the proximity of a design to a feasible boundary may be related to the reliability and robustness of the design in the face of variation in manufacture and use. A chemical engineering example is used to show how feasible design regions can be modelled using integer lattices and kriging methods. An algorithm using Hilbert bases is developed and used to select points on the boundary of the feasible region with arbitrary closeness so that the properties of the kriging model are such that the model remains stable and the star-shaped necessary condition is met.

Fractal interpolants on the unit circle

A methodology based on fractal interpolation functions is used in this work to define new real maps on the circle generalizing the classical ones. A partition on the circle and a scale vector enable the modification of the definition and properties of the standard periodic functions. The fractal analogues can be constructed even if the originals are not continuous. The new functions provide Hilbert bases for the square integrable maps on the circle.

Toric statistical models: parametric and binomial representations

Toric models have been recently introduced in the analysis of statistical models for categorical data. The main improvement with respect to classical log-linear models is shown to be a simple representation of structural zeros. In this paper we analyze the geometry of toric models, showing that a toric model is the disjoint union of a number of log-linear models. Moreover, we discuss the connections between the parametric and algebraic representations. The notion of Hilbert basis of a lattice is proved to allow a special representation among all possible parametrizations.

Un résultat de consistance pour des SVM fonctionnels par interpolation spline

This Note proposes a new methodology for function classification with Support Vector Machine (SVM). Rather than relying on projection on a truncated Hilbert basis as in our previous work, we use an implicit spline interpolation that allows us to compute SVM on the derivatives of the studied functions. To that end, we propose a kernel defined directly on the discretizations of the observed functions. We show that this method is universally consistent. To cite this article: N. Villa, F. Rossi, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

An undecidability result for AGh

We present an undecidability result for the verification of security protocols. Since the perfect cryptography assumption is unrealistic for cryptographic primitives with visible algebraic properties, several recent works relax this assumption, allowing the intruder to exploit these properties. We are interested in the Abelian groups theory in combination with the homomorphism axiom. We show that the security problem for a bounded number of sessions (expressed by satisfiability of symbolic deductibility constraints) is undecidable, obtaining in this way the first undecidability result concerning a theory for which unification is known to be decidable [F. Baader, Unification in commutative theories, Hilbert's basis theorem, and Gröbner bases, J. ACM 40(3) (1993) 477–503].