In this paper we present algorithms to compute finite state automata which, given any rational language, recognize the languages of normal words and n-chains. We also show how these automata can be used to compute the Hilbert series and Poincaré series for any algebra with a rational set of leading words of its minimal Gröbner basis.

The determination of a set of geometric entities that satisfy a series of geometric relations (constraints) constitutes the Geometric Constraint Satisfaction or Scene Feasibility (GCS/SF) problem. This problem appears in diierent forms in Assembly Planning, Constraint Driven Design, Computer Vision, etc. Its solution is related to the existence of roots to systems of polynomial equations. Previ...

Algebraic cryptanalysis usually requires to recover the secret key by solving polynomial equations. Faugère’s F4 is a well-known Gröbner bases algorithm to solve this problem. However, a serious drawback exists in the Gröbner bases based algebraic attacks, namely, any information won’t be got if we couldn’t work out the Gröbner bases of the polynomial equations system. In this paper, we in-dept...

We provide a polynomial time algorithm for computing the universal Gröbner basis of any polynomial ideal having a finite set of common zeros in fixed number of variables. One ingredient of our algorithm is an effective construction of the state polyhedron of any member of the Hilbert scheme Hilbn of n-long d-variate ideals, enabled by introducing the Hilbert zonotope Hn and showing that it simu...

It is well-known that one of open problems in the theory Leibniz algebras to find asuitable generalization Lie’s third theorem which associates a (local) Lie group any Liealgebra, real or complex. turns out, this related finding an appropriate analogue Liegroup for algebras. Using notion digroup, Kinyon obtained partial solution ofthis problem, namely, class so-called split Leibnizalgebras. A d...

Bracket coefficients for polynomials are introduced. These are like specific precision floating point numbers together with error terms. Working in terms of bracket coefficients, an algorithm that computes a Grobner basis with floating point coefficients is presented, and a new criterion for determining whether a bracket coefficient is zero is proposed. Given a finite set F of polynomials with ...

Grabner basis is olle of the most important tools ill recent symbolic algebraic computations. Howe\'er, computing a Grobncr basis fo r the given polynomial ideal is lIot easy and it is lIot numcric.. . l1y stable if polynomials have inexact coefficients. I n this paper, we study what we should get for computing a Grabner basis with inexact coefficients alld introduce a naive method to compute a...

This is the first of two articles devoted to a comprehensive exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper is entirely devoted to the study of the tensor-product (infinite-level) limit of fusions rules. We consider thus in detail the problem of constructing tensor-product generating functions in finite Lie algebras. From the be...