Computing Grobner Bases Research Articles

On polynomial curves in the affine plane

A curve that can be parametrized by polynomials is called a polynomial curve. It is well-known that a polynomial curve has only one place at infinity. Let $C$ be a curve with one place at infinity. Sathaye presented the following question raised by Abhyankar: Is there a polynomial curve associated with the semigroup generated by pole orders of $C$ at infinity? In this paper, we give a negative answer to this question using Grobner basis computation.

Minimal invariant Markov basis for sampling contingency tables with fixed marginals

In this paper we define an invariant Markov basis for a connected Markov chain over the set of contingency tables with fixed marginals and derive some characterizations of minimality of the invariant basis. We also give a necessary and sufficient condition for uniqueness of minimal invariant Markov bases. By considering the invariance, Markov bases can be presented very concisely. As an example, we present minimal invariant Markov bases for all 2 × 2 × 2 × 2 hierarchical models. The invariance here refers to permutation of indices of each axis of the contingency tables. If the categories of each axis do not have any order relations among them, it is natural to consider the action of the symmetric group on each axis of the contingency table. A general algebraic algorithm for obtaining a Markov basis was given by Diaconis and Sturmfels (The Annals of Statistics, 26, 363–397, 1998). Their algorithm is based on computing Grobner basis of a well-specified polynomial ideal. However, the reduced Grobner basis depends on the particular term order and is not symmetric. Therefore, it is of interest to consider the properties of invariant Markov basis.

Parallel computation of Janet and Gr�bner bases over rational numbers

In this paper, a parallel algorithm for computation of polynomial Janet bases is considered. After construction of a Janet basis, a reduced Grobner basis (which is a subset of the Janet basis) is extracted from it without any additional reductions. The algorithm discussed is an improved version of an earlier suggested parallel algorithm. The efficiency of a C implementation of the algorithm and its scalability are illustrated by way of the standard test examples that are often used for comparing various algorithms and codes for computing Grobner bases.

Some characterizations of minimal Markov basis for sampling from discrete conditional distributions

In this paper we given some basic characterizations of minimal Markov basis for a connected Markov chain, which is used for performing exact tests in discrete exponential families given a sufficient statistic. We also give a necessary and sufficient condition for uniqueness of minimal Markov basis. A general algebraic algorithm for constructing a connected Markov chain was given by Diaconis and Sturmfels (1998,The Annals of Statistics,26, 363–397). Their algorithm is based on computing Grobner basis for a certain ideal in a polynomial ring, which can be carried out by using available computer algebra packages. However structure and interpretation of Grobner basis produced by the packages are sometimes not clear, due to the lack of symmetry and minimality in Grobner basis computation. Our approach clarifies partially ordered structure of minimal Markov basis.

The Chen-Reed-Helleseth-Truong Decoding Algorithm and the Gianni-Kalkbrenner Gröbner Shape Theorem

In a paper from Chen, Reed, Helleseth and Truong, [10] (cf. also Loustaunau and York [10]) Grobner bases are applied as a preprocessing tool in order to devise an algorithm for decoding a cyclic code over GF(q) of length n. The Grobner basis computation of a suitable ideal allows us to produce two finite ordered lists of polynomials over GF(q),

Optimal design of synthesis filters in multidimensional perfect reconstruction FIR filter banks using Grobner bases

For a given low-pass analysis filter in an n-dimensional (n-D) perfect reconstruction (PR) FIR filter bank, there exists certain degree of freedom in designing a synthesis filter to make the overall filter bank have PR property, and Quillen-Suslin Theorem can be used to precisely determine the degree of freedom. Designing a PR synthesis filter with optimal frequency response amounts to minimizing the deviation between the frequency responses of the synthesis filter and an ideal filter under the PR constraint. In this paper, a method based on Grobner bases computation is developed for optimally designing critically sampled n-D PR FIR filter banks when one analysis filter is known. This design method can be easily combined with various existing design methods including minimax optimization and weighted least squares optimization, and can incorporate additional design goals including linear phase and bandpass characteristic.

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.

Parallel computer algebra software as a Web component

In the field of computer algebra, joining separate computer algebra systems with individual strengths into problem solving environments is an important recent research direction. In this paper we describe how we wrapped a specialized computer algebra system, which uses shared-memory parallelism to speed up Grobner basis computations, into a Web component that can be used by other systems. By this component also a new algorithmic development in Grobner basis computations – the use of the Fractal Grobner Walk – was made available to various general purpose computer algebra systems in the form of a single optimized implementation. Previously, all of the important algorithmic improvements to an algorithm had to be reimplemented in the general purpose systems. Our implementation allows access to the Grobner basis computations via ‘intelligent’ graphical user interfaces and via scripting from existing general purpose algebra systems. © 1998 John Wiley & Sons, Ltd.

Semi-Invariants, equivariants and algorithms

The results from Invariant Theory and the results for semi-invariants and equivariants are summarized in a suitable way for combining with Grobner basis computation. An algorithm for the determination of fundamental equivariants using projections and a Poincare series is described. Secondly, an algorithm is given for the representation of an equivariant in terms of the fundamental equivariants. Several ways for the exact determination of zeros of equivariant systems are discussed.

Distributed data structures and algorithms for Gr�bner basis computation

We present the design and implementation of a parallel algorithm for computing Grobner bases on distributed memory multiprocessors. The parallel algorithm is irregular both in space and time: the data structures are dynamic pointer-based structures and the computations on the structures have unpredictable duration. The algorithm is presented as a series of refinements on atransition rule program, in which computation proceeds by nondeterministic invocations of guarded commands. Two key data structures, a set and a priority queue, are distributed across processors in the parallel algorithm. The data structures are designed for high throughput and latency tolerance, as appropriate for distributed memory machines. The programming style represents a compromise between shared-memory and message-passing models. The distributed nature of the data structures shows through their interface in that the semantics are weaker than with shared atomic objects, but they still provide a shared abstraction that can be used for reasoning about program correctness. In the data structure design there is a classic trade-off between locality and load balance. We argue that this is best solved by designing scheduling structures in tandem with the state data structures, since the decision to replicate or partition state affects the overhead of dynamically moving tasks.

Computing minimal polynomials and the degree of unfaithfulness

We show how to compute, based on the Grobner basis computation, the minimal polynomial of n polynomials in n- 1 variables which contain n- 1 algebraically independent elements. We apply this result to automorphisms of polynomial rings.We also generalize these to rational functions and obtain a criterion for a rational map of an affne space to be birational.

An algorithm for determining the size of symmetry groups

To determine the symmetry group of pointor Lie-symmetries of a differential equation is of great theoretical and practical importance, in particular for determining closed form solutions. There does not seem to exist an algorithm that finds this group in general. However, it is always possible to determine thesize of the symmetry group. In this article an algorithm is described that determines for any system of algebraic partial differential equations the number of parameters if the symmetry group is finite, and the number of unspecified functions and its arguments if it is infinite. To this end the so calleddetermining system is transformed into aninvolutive system by means of a critical-pair/completion algorithm similar like it is applied for computing Grobner bases in polynomial ideal theory. The foundation for obtaining this form is the theory of Riquier and Janet for partial differential equations. The algorithmInvolution System has been implemented in several computer algebra systems as part of the packageSPDE. Various results that have been obtained by applying it are presented as well. If symmetry analysis is considered as part of the more general process of obtaining the best possible information on the solutions of a differential equation, the algorithm described in this article removes the heuristics which is usually involved in making the transition from analytical to numerical methods.

Extended parallelism in the Gr�bner basis algorithm

This paper presents a new parallel implementation to compute Grobner bases utilizing two different forms of parallelism. A coarse-grain technique developed by Jean-Phillipe Vidal expands and reducesS-polynomials in parallel. A finegrain technique, proposed by Melenk and Neun, constructs a pipeline of processors to overlap execution of the reduction operations. A hybrid algorithm that outperforms both of the original approaches is presented. The combined algorithm requires the user to select the appropriate allocation of processors to the two styles of parallelism, and uses this static assignment throughout the computation. The paper also discusses the design and implementation approaches used to construct an efficient version of this algorithm.

Procedures to investigate injectivity of polynomial maps and to compute the inverse

In this paper a polynomial map from C n to C m is studied in order to investigate if it is injective out of a set of measure zero. We propose a procedure, based on truncated Grobner basis computations, which when successful, allows to reduce the problem to an easier map, and so gives a speed-up of the general algorithms using Grobner basis techniques. Moreover, for the special case of a polynomial map from C n to C n where the polynomials are at most quadratic, we propose two criteria for non-injectivity based on the structure of the Jacobian matrix and requiring only basic symbolic computations.