This paper addresses the critical issue of the development of intelligent or assisting tools for the scientist who is working in the initial problem formulation and mathematical model representation stage of research. In particular, examples of that representation in fluid dynamics and instability theory are discussed. The creation of a mathematical model that is ready for application of certai...

In this paper, we propose the concept of involutive direction ss a vector representation for the concept of involutive division proposed by Gerdt and hi co-workers. With this representation, most of the properties of involutive divisions such as Noetherity, Artinity, and constructivity, can be greatly simplified. A new algorithm to compute the involutive completion is also given. Based on the v...

In this paper, we consider a monomial ideal J / P := A[x1, . . . , xn], over a commutative ring A, and we face the problem of the characterization for the familyMf (J) of all homogeneous ideals I / P such that the A-module P/I is free with basis given by the set of terms in the Gröbner escalier N(J) of J. This family is in general wider than that of the ideals having J as initial ideal w.r.t. a...

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.

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...

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...