Grobner Basis Theory Research Articles

Characteristic Modules of Dual Extensions and Gröbner Bases

Let C be a finite dimensional directed algebra over an algebraically closed field k and A = A(C) the dual extension of C. The characteristic modules of A are constructed explicitly for a class of directed algebras, which generalizes the results of Xi. Furthermore, it is shown that the characteristic modules of dual extensions of a certain class of directed algebras admit the left Grobner basis theory in the sense of E. L. Green.

Algorithm and architecture for a Galois field multiplicative arithmetic processor

We present a new algorithm for generic multiplicative computations of the form ab/c in GF(p/sup m/), including multiplication, inversion, squaring, and division. The algorithm is based on solving a sequence of congruences that are derived from the theory of Grobner bases in modules over the polynomial ring GF(p)[x]. Its corresponding hardware and software architectures can be successfully used in applications such as error control coding and cryptography. We describe a versatile circuit associated with the algorithm for the most important case p=2. The same hardware can be used for a range of field sizes thus permitting applications in which different levels of error control or of security are required by different classes of user. The operations listed are all performed by the hardware in the same number of clock cycles, which prevents certain side-channel attacks. The loss in performance by having 2m iterations for multiplication is compensated by the full parameterization of the Galois field and the ability to perform division and multiplication in parallel.

A Constructive Solution to Interconnection and Decomposition Problems with Multidimensional Behaviors

By a multidimensional behavior, we mean the solution space of a linear constant-coefficient system of partial difference or differential equations. Within the behavioral framework, a natural concept of interconnection has been introduced by J. C. Willems. The regular interconnection problem can be formulated as follows: Given a behavior (the plant), find---if possible---another behavior (a controller) such that their interconnection is not redundant and is equal to a certain desired behavior. We give a constructive solution to this problem, which is in terms of polynomial matrix equations that can be solved using the theory of Grobner bases. We also study a dual problem concerning the existence of direct sum decompositions of multidimensional behaviors. Finally, we present a unified framework for studying both problems as special cases of constructing relative complements in the lattice of behaviors.

Quasi-Steady State Kinetics of Simple Sequential Multienzyme Reactions with Single Substrates

Applications of computer algebra technique to kinetic analysis of multienzyme reaction including two and three distinct enzymes were described in the literature, which mainly depend on Grobner Basis theory. In the present study, we have applied the same methodology to a hypothetical system with four enzymes. After deriving a composite rate law for a reaction system with four distinct enzymes using the computer algebra system MAPLE, the composite rate law was fitted to simulated data for time versus product concentration to yield estimates of the kinetic parameters.

Problems and algorithms for affine semigroups

Affine semigroups—discrete analogues of convex polyhedral cones—mark the cross-roads of algebraic geometry, commutative algebra and integer programming. They constitute the combinatorial background for the theory of toric varieties, which is their main link to algebraic geometry. Initiated by the work of Demazure [17] and Kempf, Knudsen, Mumford and Saint-Donat [33] in the early 70s, toric geometry is still a very active area of research. However, the last decade has clearly witnessed the extensive study of affine semigroups from the other two perspectives. No doubt, this is due to the tremendously increased computational power in algebraic geometry, implemented through the theory of Grobner bases, and, of course, to the modern computers. In this article we overview those aspects of this development that have been relevant for our own research, and pose several open problems. Answers to these problems would contribute substantially to the theory. The paper treats two main topics: (1) affine semigroups and several covering properties for them and (2) algebraic properties for the corresponding rings (Koszul, Cohen-Macaulay, different “sizes” of the defining binomial ideals). We emphasize the special case when the initial data are encoded into lattice polytopes. The related objects—polytopal semigroups and algebras— provide a link with the classical theme of triangulations into unimodular simplices. We have also included an algorithm for checking the semigroup covering property in the most general setting (Section 4). Our counterexample to certain covering conjectures (Section 3) was found by the application of a small part of this algorithm. The general algorithm could be used for a deeper study of affine semigroups.

Applications of the theory of Gröbner bases to the study of linear recurring arrays

This is a small survey of applications of the theory of Grobner bases to the study of linear recurring arrays. It applies some properties of Grobner bases to studying linear recurring arrays and contains recent new results on linear recurring arrays.

Strong Gröbner bases and cyclic codes over a finite-chain ring.

Abstract We present a unified theory of Grobner bases over a principal ideal ring with a view to applications in Coding Theory, for example to the structure of codes over a finite-chain ring, e.g. a Galois ring. Our theory is effective and yields an extension of Buchberger's algorithm to principal ideal rings. In particular, given a set of generators for a cyclic code over a Galois ring, we can compute a ‘minimal strong’ Grobner basis for the code. We give some applications to cyclic codes.

On the existence of order functions

The notions of well-behaving sequences and order functions is fundamental in the elementary treatment of geometric Goppa codes. The existence of order functions is proved with the theory of Grobner bases.

On the Finiteness of Gröbner Bases Computation in Quotients of the Free Algebra

We investigate, for quotients of the non-commutative polynomial ring, a property that implies finiteness of Grobner bases computation, and examine its connection with Noetherianity. We propose a Grobner bases theory for our factor algebras, of particular interest for one-sided ideals, and show a few applications, e.g. how to compute (one-sided) syzygy modules.

Minimised Geometric Buchberger Algorithm for Integer Programming

Recently, various algebraic integer programming (IP) solvers have been proposed based on the theory of Grobner bases. The main difficulty of these solvers is the size of the Grobner bases generated. In algorithms proposed so far, large Grobner bases are generated by either introducing additional variables or by considering the generic IP problem IPA,C. Some improvements have been proposed such as Hosten and Sturmfels' method (GRIN) designed to avoid additional variables and Thomas' truncated Grobner basis method which computes the reduced Grobner basis for a specific IP problem IPA,C(b) (rather than its generalisation IPA,C). In this paper we propose a new algebraic algorithm for solving IP problems. The new algorithm, called Minimised Geometric Buchberger Algorithm, combines Hosten and Sturmfels' GRIN and Thomas' truncated Grobner basis method to compute the fundamental segments of an IP problem IPA,C directly in its original space and also the truncated Grobner basis for a specific IP problem IPA,C(b). We have carried out experiments to compare this algorithm with others such as the geometric Buchberger algorithm, the truncated geometric Buchberger algorithm and the algorithm in GRIN. These experiments show that the new algorithm offers significant performance improvement.

Extension Modules in Behavioral Linear Systems Theory

For certain signal spaces, the algebraic concept of extension modules yields a full classification of system theoretic properties such as controllability and autonomy. The present paper summarizes the results of various authors on the subject, and unifies them in the language of extension modules. The computation of extension modules is based on the computation of syzygies (i.e., kernels of module homomorphisms). This can be done constructively, using the theory of Grobner bases.

Gröbner Bases for Problem Solving in Multidimensional Systems

The objective here is to underscore recent usage of the algorithmic theory of Grobner bases in multidimensional systems since that possibility was highlighted about fifteen years back. The main contribution here focuses on the constructive aspects of the solution, known to exist, of the two-band multidimensional IIR perfect reconstruction problem using Grobner bases. Other recent research results on the subject with future prospects are also briefly cited.

Gröbner Bases and Systems Theory

We present the basic concepts and results of Grobner bases theory for readers working or interested in systems theory. The concepts and methods of Grobner bases theory are presented by examples. No prerequisites, except some notions of elementary mathematics, are necessary for reading this paper. The two main properties of Grobner bases, the elimination property and the linear independence property, are explained. Most of the many applications of Grobner bases theory, in particular applications in systems theory, hinge on these two properties. Also, an algorithm based on Grobner bases for computing complete systems of solutions (“syzygies”) for linear diophantine equations with multivariate polynomial coefficients is described. Many fundamental problems of systems theory can be reduced to the problem of syzygies computation.

Footprints or generalized Bezout's theorem

In two previous papers, the first by Feng, Rao, Berg, and Zhu (see ibid., vol.43, p.1799-810, 1997) and the second by Feng, Zhu, Shi, and Rao (see Proc. 35th. Afferton Conf. Communication, Control and Computing, p.205-14, 1997), the authors use a generalization of Bezout's theorem to estimate the minimum distance and generalized Hamming weights for a class of error correcting codes obtained by evaluation of polynomials in points of an algebraic curve. The main aim of this article is to show that instead of using this rather complex method the same results and some improvements can be obtained by using the so-called footprint from Grobner basis theory. We also develop the theory further such that the minimum distance and the generalized Hamming weights not only can be estimated but also can actually be determined.

Deformations of Codimension 2 Toric Varieties

We prove Sturmfels' conjecture that toric varieties of codimension two have no other flat deformations than those obtained by Grobner basis theory.

The application of computational algebraic geometry to the analysis of designed experiments: a case study

Computational algebraic geometry can be used to solve estimability/identifiability problems in the design of experiments. The key is to replace the design as a set of points by the polynomials whose solutions are the design points. The theory and application of Grobner bases allows one to find a unique saturated model for each so-called monomial ordering of the independent factors. A case study in engine mapping is fully worked out and employs a simple plotting method for modelling.

Gröbner Bases for Ideals in Laurent Polynomial Rings and their Application to Systems of Difference Equations

We develop a basic theory of Grobner bases for ideals in the algebra of Laurent polynomials (and, more generally, in its monomial subalgebras). For this we have to generalize the notion of term order. The theory is applied to systems of linear partial difference equations (with constant coefficients) on ℤn. Furthermore, we present a method to compute the intersection of an ideal in the algebra of Laurent polynomials with the subalgebra of all polynomials.

Cohen–Macaulay Rings Associated with Digraphs

Let G be a digraph, that is, a pair of sets consisting of a set of vertices Ž . and a set of directed edges for a more precise definition, see Section 3 . It is an interesting problem to know how to count the Hamilton cycles of G, that is, cycles containing all vertices of G. In this paper, we will give the upper bound of the number of them by using the theory of commutative rings. It is natural to assume that G is strongly connected, otherwise there is no Hamilton cycle. The way to give it is to associate G with a Cohen]Macaulay ring which is positively graded of dimension 1 and to compute its Macaulay type. In fact, the associated ring is a monoid ring, r Ž which is defined for a submodule of Z where Z is the ring of integers for . the definition, see Section 1 . For this reason, we will compute the Macaulay type of Cohen]Macaulay monoid ring in a special case in Section 2. w x Ž . Let A s k X , . . . , X k is a field . In Section 1, we define the 1 r Ž . r Ž . polynomial F for g Z and the ideal I V of A for a submodule V of r Ž . Ž . Z for the definition, see Section 1 . And we call the ring ArI V a monoid ring. These definitions and a few properties of this ring are found w x in 5 . The monoid ring is always a graded ring. If it is positively graded, we can use the theory of Grobner bases and show a basic lemma about Ž . Ž . generation of elements in its defining ideal I V see Proposition 1.3 . In Section 2, we only treat a special case: V is a submodule generated by r Ž . Ž . , . . . , in Z with s ) 0, s F 0 for each j / i and q ??? q 1 r i i i j 1 r s 0, where s denotes the ith entry. Further assume that the monoid ring i Ž . ArI V is positively graded of dimension 1. We will see in the next section that these conditions are satisfied if V is associated with a strongly connected digraph. Thus to investigate this case makes sense. Under these Ž . conditions ArI V is Cohen]Macaulay and we will compute its Macaulay type.

Decoding Affine Variety Codes Using Gröbner Bases

We define a class of codes that we call affine variety codes. These codes are obtained by evaluating functions in the coordinate ring of an affine variety on all the F_q-rational points of the variety. We show that one can, at least in theory, decode these codes up to half the true minimum distance by using the theory of Grobner bases. We extend results of A. B. Cooper and of X. Chen, I. S. Reed, T. Helleseth, and T. K. Truong.

Boolean Term Orders and the Root System B n

A Boolean term order is a total order on subsets of [n] ={1,..., n} such that ∅ ≺ α for all α $$ \subseteq$$ [n], α ≠ ∅, and α ≺ β ⇒ α ∪ γ ≺ β ∪ γ for all γ with γ ∩(α ∪ β) = ∅. Boolean term orders arise in several different areas of mathematics, including Grobner basis theory for the exterior algebra, and comparative probability. The main result of this paper is that Boolean term orders correspond to one-element extensions of the oriented matroid M(Bn), where Bn is the root system {ei : 1 ≤ i ≤ n} ∪{ei ± ej : 1 ≤ i < j ≤ n}. This establishes Boolean term orders in the framework of the Baues problem, in the sense of (Reiner, 1998). We also define a notion of coherence for a Boolean term order, and a flip relation between different term orders. Other results include examples of noncoherent term orders, including an example exhibiting flip deficiency, and enumeration of Boolean term orders for small values of n.