In this paper, we give a brief overview on Gröbner bases theory, addressed to novices without prior knowledge in the field. After explaining the general strategy for solving problems via the Gröbner approach, we develop the concept of Gröbner bases by studying uniquenss of polynomial division ("reduction"). For explicitly constructing Gröbner bases, the crucial notion of S polynomials is introd...

This paper describes and analyzes a method for computing border bases of a zero-dimensional ideal I . The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending the one provided in [29]. This general border basis algorithm weakens the monomial ordering requirement for Gröbner bases computations. It is up to date the ...

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

Interpolation is a prime tool in algebraic computation while symmetry qualitative feature that can be more relevant to mathematical model than the numerical accuracy of parameters. The article shows how exactly preserve multivariate interpolation exploiting it alleviate computational cost. We revisit minimal degree and least with adapted bases, rather monomial bases. For space linear forms inva...

We focus on Gröbner bases for modules of univariate polynomial vectors over a ring. We identify a useful property, the " predictable leading monomial (PLM) property " that is shared by minimal Gröbner bases of modules in F[x] q , no matter what positional term order is used. The PLM property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable deg...

We give a new realization of crystal bases for finite dimensional irreducible modules over special linear Lie algebras using the monomials introduced by H. Nakajima. We also discuss the connection between this monomial realization and the tableau realization given by Kashiwara and Nakashima. Introduction The quantum groups, which are certain deformations of the universal enveloping algebras of ...

Introduction. It is notoriously difficult to deduce anything about the structure of an ideal or scheme by directly examining its defining polynomials. A notable exception is that of monomial ideals. Combined with techniques for making fiat degenerations of arbitrary ideals into monomial ideals (typically, using Gr6bner bases), the theory of monomial ideals becomes a useful tool for studying gen...

Abstract. We show there is a natural connection between Latin squares and commutative sets of monomials defining geometric structures in finite phase-space of prime power dimensions. A complete set of such monomials defines a mutually unbiased basis (MUB) and may be associated with a complete set of mutually orthogonal Latin squares (MOLS). We translate some possible operations on the monomial ...

The interpolation step of Sudan’s list decoding of Reed-Solomon codes sets forth the problem of finding the minimal polynomial of the ideal of interpolating polynomials with respect to a certain monomial order. An efficient algorithm that solves the problem is presented based on the theory of Gröbner bases of modules. In a special case, this algorithm is shown to be equivalent with the Berlekam...

In this paper we describe an efficient involutive algorithm for constructing Gröbner bases of polynomial ideals. The algorithm is based on the concept of involutive monomial division which restricts the conventional division in a certain way. In the presented algorithm a reduced Gröbner basis is the internally fixed subset of an involutive basis, and having computed the later, the former can be...