Abstract: In construction of parametric Gr\"obner bases, we usually assume that parameters can take arbitrary values. In case, however, there exist some constraints among parameters, it is more natural to construct parametric Gr\"obner bases for only parameters satisfying such constraints. Using this idea, we formalized parametric Gr\"obner bases in terms of ACGB(Alternative Comprehensive Gr\"obner Bases) which we proposed in ISSAC2002. This natural formalization leads us to an interesting and desirable fact that is discrete comprehensive Gr\"obner bases studied by us can be naturally defined and generalized as special instances of ACGB.
Abstract: Let $K$ be a partial difference field with a basic set $\sigma$ and let a partition of $\sigma$ into $p$ disjoint subsets be fixed. We consider a generalization of the Grobner basis method to free difference modules and apply the new technique to the study of difference dimension.
Abstract: Let K be a partial difference field with a basic set S and let a partition of S into p disjoint subsets be fixed. We consider a generalization of the Gröbner basis method to free difference modules and apply the new technique to the study of difference dimension. In particular, we generalize the author's theorems on difference dimension polynomials and obtain new invariants of a finitely generated difference field extension.
Abstract: The author uses new results and techniques from the method of Groebner bases to devise an efficient and reliable algorithm for finding implicit representations of geometric objects. The process of finding implicit representations is also known as implicitization. Implicitization is an "established" research topic, which is studied in several research areas such as computer aided geometric design (CAGD), visualization and solid modeling. However, until now there is no method for implicitization that is both reliable and efficient. The implicitization method using Groebner bases is reliable but not efficient and the other two main approaches, namely Dixon's resultants and moving surfaces, are more efficient but not reliable. In the context of this research, a reliable method is a method that theoretically never fails to give a correct answer. An efficient method is a method that has a better complexity or a more reasonable running time and consumes less memory space in order to support interaction between the designer/user and the computer. In this paper, the author investigates the use of the deterministic Groebner walk method to convert a parametric representation of a surface into its implicit form. For rational parametric surfaces, the author uses a different approach to deal with base points in that the calculation of a Groebner basis for the starting cone is no longer needed. This approach will help to improve the efficiency of the algorithms because the usual calculation of the implicit representation, which often consumes a lot of time and memory space, is replaced by a sequence of small calculations along the walking path and then lift the results using linear transformations. Another important task of this research is to reduce the number of terms of the intermediate polynomials and find criteria for detecting unnecessary reduction. Experimental results with the deterministic Groebner walk conversion method show that most of the time for implicitization is used for reducing the minimal bases after lifting. Obviously, not all of the polynomials need to be reduced; for example, at the last cone most of the polynomials do not need to be reduced. Therefore, detecting all unnecessary reductions would be a leap in improving the efficiency of algorithms for implicitization. This will also significantly reduce the memory space needed for the calculation. This research will hopefully produce improved algorithms for implicitization, which will never fails, faster and consume less memory space. Such an efficient and reliable algorithm will make an impact on research areas dealing with designing curves and surfaces. For example, it can be used for finding the intersection of surfaces, to verify whether or not a point lies on a surface, etc.
Abstract: Two linear codes are permutation-equivalent if they are equal up to a permutation on the codeword coordinates. We present an invariant for the class of linear codes, i.e. a mapping such that any two permutation equivalent codes take the same value. This mapping is closely related to the ideas behind FGLM (Faugere, Gianni, Lazard and Mora 1993), mainly to the presence of linear algebra techniques into the framework of some Groebner bases tools. We show how these Groebner basis tools can be used to determine if two linear codes are equivalent or not.
Abstract: In this article the definitions of projections of involutive divisions and graphs are formulated. With their help, series of new examples of involutive divisions are constructed. The criteria of completeness and globalness of involutive divisions in the language of the graphs are formulated and proved. It is obtained and proved the criterion of noetherian global involutive division corresponds to the given graph.
Abstract: We present an algorithm that incrementally computes a Gr\"obner basis for the vanishing ideal of any finite set of points in ${\mathbb{F}}^m$ under any given monomial order, and we apply this algorithm to multivariate polynomial interpolation and rational function interpolation. Our algorithm is polynomial in the dimension, $m$, and the number of points, $n$. This approach is completely different from the several forms which rely on Gauss elimination and is, in fact, a natural generalization of univariate Newton interpolation.
Abstract: Statistical disclosure limitation applies statistical tools to address the problem of limiting sensitive information releases about individuals/groups while maintaining proper statistical inferences. Within this context, Dobra and Fienberg (2000,2002) have recently employed Gr\"obner bases in connection with graphical models given a set of marginal counts to establish bounds and distributions for cell entries in contingency tables. Building on their work and that of Garcia et. al. (2003), we explore the applicability of Gr\"obner bases in connection with Bayesian networks to address the issue of data confidentiality when the released information is in the form of an arbitrary collection of conditional and marginal distributions.
Abstract: In this talk, the author will present a new Maple package that uses the deterministic Groebner walk method for fast conversion of Groebner bases. Previous efforts to implement the Groebner walk method encountered some technical problems, which resulted in packages with many limitations. This new package overcame most of the known">Edgar Martinez-Moro (Universidad de Valladolid,Spain)
Abstract: Two linear codes are permutation-equivalent if they are equal up to a permutation on the codeword coordinates. We present an invariant for the class of linear codes, i.e. a mapping such that any two permutation equivalent codes take the same value. This mapping is closely related to the ideas behind FGLM (Faugere, Gianni, Lazard and Mora 1993), mainly to the presence of linear algebra techniques into the framework of some Groebner bases tools. We show how these Groebner basis tools can be used to determine if two linear codes are equivalent or not.
Abstract: Foveal approximation spaces were introduced by Mallat, with a view towards efficient approximation of real-valuated functions with isolated singularities. Le Pennec and Mallat then showed how to use these techniques for image compression, using bandelet bases. In this talk, we study foveal spaces generated by successive dilatations of two mother foveal wavelets. We detail the computations necessary to obtain foveal wavelets with good approximation properties, and present their practical performance. To ensure an approximation error with small amplitude, the families of wavelets under consideration must reproduce high degree polynomials, and decay fast enough at zero and infinity. In particular, for fixed d, foveal wavelets that reproduce polynomials up to degree d and have minimal support are solutions of a suitable polynomial system. We have solved these systems for a degree $d$ up to 10. Additional regularity requirements can be put on the mother wavelets, so as to ensure regularity of the approximations. Then wavelets with minimal support and minimal oscillation are solutions of non-algebraic systems. We present the performance of wavelets obtained by solving polynomial approximations of these systems. To solve the polynomial systems, we used the Kronecker package developped in Magma by Lecerf. This package offers an alternative to Groebner bases for polynomial system solving, on the basis of the work of Giusti, Heintz, Pardo and collaborators in the TERA project.