Related papers: Computing Border Bases without using a Term Orderi…
The universal Gr\"obner basis of an ideal is a Gr\"obner basis with respect to all term orders simultaneously. The aim of this paper is to present an algorithmic approach to compute the universal Gr\"obner basis for the toric ideal…
Insa and Pauer presented a basic theory of Groebner basis for differential operators with coefficients in a commutative ring in 1998, and a criterion was proposed to determine if a set of differential operators is a Groebner basis. In this…
One of the biggest open problems in computational algebra is the design of efficient algorithms for Gr{\"o}bner basis computations that take into account the sparsity of the input polynomials. We can perform such computations in the case of…
Standard noncommutative Gr\"obner basis procedures are used for computing ideals of free noncommutative polynomial rings over fields. This paper describes Gr\"obner basis procedures for one-sided ideals in finitely presented noncommutative…
Border basis schemes are open subschemes of the Hilbert scheme of $\mu$ points in an affine space $\mathbb{A}^n$. They have easily describable systems of generators of their vanishing ideals for a natural embedding into a large affine space…
We report on our experiences exploring state of the art Groebner basis computation. We investigate signature based algorithms in detail. We also introduce new practical data structures and computational techniques for use in both signature…
A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra…
We establish basic information about border rank algorithms for the matrix multiplication tensor and other tensors with symmetry. We prove that border rank algorithms for tensors with symmetry (such as matrix multiplication and the…
An algorithm to generate a minimal comprehensive Gr\"obner\, basis of a parametric polynomial system from an arbitrary faithful comprehensive Gr\"obner\, system is presented. A basis of a parametric polynomial ideal is a comprehensive…
Algorithmic computation in polynomial rings is a classical topic in mathematics. However, little attention has been given to the case of rings with an infinite number of variables until recently when theoretical efforts have made possible…
Given a sequence of related modules $M_n$ defined over a sequence of related polynomial rings, one may ask how to simultaneously compute a finite Gr\"obner basis for each $M_n$. Furthermore, one may ask how to simultaneously compute the…
We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a…
Developed by Buchberger for commutative polynomial rings, Groebner Bases are frequently applied to solve algorithmic problems, such as the congruence problem for ideals. Until now, these ideas have been transmitted to different in part…
One studies a particular algebraic system where the unknowns are matrices. We solve this system according to the parameters values thanks to the theory of Grobner basis.
This paper studies the concept and the computation of approximately vanishing ideals of a finite set of data points. By data points, we mean that the points contain some uncertainty, which is a key motivation for the approximate treatment.…
A contemporary and exciting application of Groebner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to…
We explain an algorithm for finding a boundary link Seifert matrix for a given Alexander polynomial. The algorithm depends on several choices and therefore makes it possible to find non-equivalent Seifert matrices for a given Alexander…
Let $I_1\subset I_2\subset\dots$ be an increasing sequence of ideals of the ring $\Bbb Z[X]$, $X=(x_1,\dots,x_n)$ and let $I$ be their union. We propose an algorithm to compute the Gr\"obner base of $I$ under the assumption that the…
In this paper we consider systems of partial (multidimensional) linear difference equations. Specifically, such systems arise in scientific computing under discretization of linear partial differential equations and in computational high…
Experiences with the implementation of strong Gr\"obner bases respectively standard bases for polynomial rings over principal ideal rings are explained: different strategies for creating the pair set, methods to avoid coefficient growth and…