Minimal Involutive Bases

We simplify an earlier paper of the same title by not using syzygy polynomials and by not using a trichotomy of inverse forms. Let $\K$ be a field and $\M=\K[x^{-1},z^{-1}]$ denote Macaulay's $\K[x,z]$ module of inverse polynomials; here $z$ and $z^{-1}$ are homogenising variables. An inverse form $F\in\M$ has a homogeneous annihilator ideal, $\I_F$\,. In an earlier paper we inductively constructed an ordered pair ($f_1$\,,\,$f_2$) of forms in $\K[x,z]$ which generate $\I_F$....

In this paper we present a version of the general polynomial involutive algorithm for computing Janet bases specialized to toric ideals. The relevant data structures are Janet trees which provide a very fast search for a Janet divisor. We broach also efficiency issues in view of application of the algorithm presented to computation of toric ideals.

Let T(x) in k[x] be a monic non-constant polynomial and write R=k[x] / (T) the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) in R[y]. In a first part, T = p^e is assumed to be the power of an irreducible polynomial p. A new algorithm that computes a minimal lexicographic Groebner basis of the ideal ( a, b, p^e), is introduced. A second part extends this algorithm when T is general through the "local/global" principle realized by a generalization of "dynam...

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I. It is well known that minimal polynomials may be computed via elimination, therefore this is considered to be a "resolved problem". But being the key of so many computations, it is worth investigating its meaning, its optimization, its a...

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 based Groebner basis algorithms and more traditional variations of the classic Buchberger algorithm. Our conclusions are based on experiments using our new freely available open source standalone C++ library.

We investigate the use of noncommutative Groebner bases in solving partially prescribed matrix inverse completion problems. The types of problems considered here are similar to those in [BLJW]. There the authors gave necessary and sufficient conditions for the solution of a two by two block matrix completion problem. Our approach is quite different from theirs and relies on symbolic computer algebra. Here we describe a general method by which all block matrix completion pro...

In this paper the relation between Pommaret and Janet bases of polynomial ideals is studied. It is proved that if an ideal has a finite Pommaret basis then the latter is a minimal Janet basis. An improved version of the related algorithm for computation of Janet bases, initially designed by Zharkov, is described. For an ideal with a finite Pommaret basis, the algorithm computes this basis. Otherwise, the algorithm computes a Janet basis which need not be minimal. The obtained...

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 [MT'05]. This general border basis algorithm weakens the monomial ordering requirement for \grob bases computations. It is up to date the most general setting for representing quotient algebras, embedding into a single ...

This text consists of five relatively systematic notes on Gr\"obner bases and free resolutions of modules over solvable polynomial algebras.

In this paper we introduce a working generalization of the theory of Gr\"obner bases for algebras of partial difference polynomials with constant coefficients. One obtains symbolic (formal) computation for systems of linear or non-linear partial difference equations arising, for instance, as discrete models or by the discretization of systems of differential equations. From an algebraic viewpoint, the algebras of partial difference polynomials are free objects in the category...