Computing a Gröbner basis of a polynomial ideal over a Euclidean domain

AbstractAn algorithm for computing a Gröbner basis of a polynomial ideal over a Euclidean domain is presented. The algorithm takes an ideal specified by a finite set of polynomials as its input; it produces another finite basis of the same ideal with the properties that using this basis, every polynomial in the ideal reduces to 0 and every polynomial in the polynomial ring reduces to a unique normal form. The algorithm is an extension of Buchberger's algorithms for computing Gröbner bases of polynomial ideals over an arbitrary field and over the integers as well as our algorithms for computing Gröbner bases of polynomial ideals over the integers and the Gaussian integers. The algorithm is simpler than other algorithms for polynomial ideals over a Euclidean domain reported in the literature; it is based on a natural way of simplifying polynomials by another polynomial using Euclid's division algorithm on the coefficients in polynomials. The algorithm is illustrated by showing how to compute Gröbner bases for polynomial ideals over the integers, the Gaussian integers as well as over algebraic integers in quadratic number fields admitting a division algorithm. A general theorem exhibiting the uniqueness of a reduced Gröbner basis of an ideal, determined by an admissible ordering on terms (power products) and other conditions, is discussed

Generalization of the Lee-O'Sullivan List Decoding for One-Point AG Codes

We generalize the list decoding algorithm for Hermitian codes proposed by Lee and O'Sullivan based on Gr\"obner bases to general one-point AG codes, under an assumption weaker than one used by Beelen and Brander. Our generalization enables us to apply the fast algorithm to compute a Gr\"obner basis of a module proposed by Lee and O'Sullivan, which was not possible in another generalization by Lax.Comment: article.cls, 14 pages, no figure. The order of authors was changed. To appear in Journal of Symbolic Computation. This is an extended journal paper version of our earlier conference paper arXiv:1201.624

List Decoding Algorithm based on Voting in Groebner Bases for General One-Point AG Codes

We generalize the unique decoding algorithm for one-point AG codes over the Miura-Kamiya Cab curves proposed by Lee, Bras-Amor\'os and O'Sullivan (2012) to general one-point AG codes, without any assumption. We also extend their unique decoding algorithm to list decoding, modify it so that it can be used with the Feng-Rao improved code construction, prove equality between its error correcting capability and half the minimum distance lower bound by Andersen and Geil (2008) that has not been done in the original proposal except for one-point Hermitian codes, remove the unnecessary computational steps so that it can run faster, and analyze its computational complexity in terms of multiplications and divisions in the finite field. As a unique decoding algorithm, the proposed one is empirically and theoretically as fast as the BMS algorithm for one-point Hermitian codes. As a list decoding algorithm, extensive experiments suggest that it can be much faster for many moderate size/usual inputs than the algorithm by Beelen and Brander (2010). It should be noted that as a list decoding algorithm the proposed method seems to have exponential worst-case computational complexity while the previous proposals (Beelen and Brander, 2010; Guruswami and Sudan, 1999) have polynomial ones, and that the proposed method is expected to be slower than the previous proposals for very large/special inputs.Comment: Accepted for publication in J. Symbolic Computation. LaTeX2e article.cls, 42 pages, 4 tables, no figures. Ver. 6 added an illustrative example of the algorithm executio

A Tropical F5 algorithm

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. While generalizing the classical theory of Gr{\"o}bner bases, it is not clear how modern algorithms for computing Gr{\"o}bner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faug{\`e}re. In this article, we prove that, for homogeneous ideals, it can be adapted to the tropical case. We prove termination and correctness. Because of the use of the valuation, the theory of tropical Gr{\"o}b-ner bases is promising for stable computations over polynomial rings over a p-adic field. We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical F5 algorithm

Determinantal sets, singularities and application to optimal control in medical imagery

Control theory has recently been involved in the field of nuclear magnetic resonance imagery. The goal is to control the magnetic field optimally in order to improve the contrast between two biological matters on the pictures. Geometric optimal control leads us here to analyze mero-morphic vector fields depending upon physical parameters , and having their singularities defined by a deter-minantal variety. The involved matrix has polynomial entries with respect to both the state variables and the parameters. Taking into account the physical constraints of the problem, one needs to classify, with respect to the parameters, the number of real singularities lying in some prescribed semi-algebraic set. We develop a dedicated algorithm for real root classification of the singularities of the rank defects of a polynomial matrix, cut with a given semi-algebraic set. The algorithm works under some genericity assumptions which are easy to check. These assumptions are not so restrictive and are satisfied in the aforementioned application. As more general strategies for real root classification do, our algorithm needs to compute the critical loci of some maps, intersections with the boundary of the semi-algebraic domain, etc. In order to compute these objects, the determinantal structure is exploited through a stratifi-cation by the rank of the polynomial matrix. This speeds up the computations by a factor 100. Furthermore, our implementation is able to solve the application in medical imagery, which was out of reach of more general algorithms for real root classification. For instance, computational results show that the contrast problem where one of the matters is water is partitioned into three distinct classes

Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories

The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these short-comings. Its so-called 'embarrassing parallelizability' allows us to solve many problems and extract physical information which elude the symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.Comment: 36 page