The method of Gröbner bases, introduced by Bruno Buchberger in his thesis (1965), have become a powerful tool for constructive problems in polynomial ideal theory and related domains. Generalizations of the basic ideas to the non-commutative setting was done, as an theoretical instrument, by Bokut (1976) and Bergman (1978). From the constructive point of view, the non-commutative version of Buchberger's algorithm was presented by Mora (1986). For some special classes of non-commutative rings, Gröbner bases has been studied in more detail, e.g. solvable algebras by Kandri-Rody and Weispfenning (1990).

As the title indicates, we will here consider Gröbner bases in non-commutative polynomial rings, i.e. free associative algebras (over some field). Most of the results are just easy generalizations of the theory of Gröbner basis in commutative polynomial rings, which can be found e.g. in the textbook by Adams and Loustaunau (1994), or in the original paper by Buchberger (1985).

Our goal in this tutorial is to give a quick overview of generic initial ideals. A more comprehensive treatment is in [Gr96]. We first lay out the basic facts and notations, and then deal with a few of the more interesting points in a question and answer format.

We would like to thank Bruno Buchberger for inviting us to contribute this paper, and also for having, through his fundamental contributions, made the work discussed in this tutorial possible.

For this tutorial we let S = C[x1,…, xn]. Some of what we do also works in characteristic p > 0, but the combinatorial properties of Borel fixed monomial ideals are more complicated. Later we give an indication of what is true in this case.

Let I be a homogeneous ideal in the polynomial ring S, and choose a monomial order. Throughout this tutorial, the only monomial orders that we consider satisfy x1 > x2 > … > xn. Any such order will do, but the most interesting from our present point of view are the lexicographic and reverse lexicographic orders. Given this monomial order, we may compute a Gröbner basis {g1,…, gr} of the ideal I, using Buchberger's algorithm. The initial ideal in (I) is the monomial ideal generated by the lead terms of g1,…,gr. This monomial ideal has the same Hilbert function as I.

Introduction

Buchberger's algorithm (Buchberger, 1965; Buchberger, 1985) for the computation of Gröbner bases has became one of the most important algorithms in providing exact solutions of scientific problems in multivariate polynomial ideal theory, elimination theory and so on.

Introduction

The subject of this article are systems of linear homogeneous partial differential equations (pde's) of various kinds. Above all such equations are characterized by the number m of dependent and the number n of independent variables. Additional quantities of interest are the number of equations, the order of the highest derivatives that may occur and the function field in which the coefficients are contained. Without further specification it is the field of rational functions in the independent variables. The basic new concept to be considered in this article is the Janet base. This term is chosen because the French mathematician Maurice Janet (Janet 1920) recognized its importance and described an algorithm for obtaining it. After it had been forgotten for about fifty years, it was rediscovered (Schwarz 1992) and utilized in various applications as it is described later on.

The theory of systems of linear homogeneous pde's is of interest for its own right, independent of its applications e. g. for finding symmetries and invariants of differential equations. Any such system may be written in infinitely many ways by linearly combining its members or derivatives thereof without changing its solution set. In general it is a difficult question whether there exist nontrivial solutions at all, or what the degree of arbitrariness of the general solution is. It may be a set of constants, or a set of functions depending on a differing number of arguments.

1. In this paper, I will describe some recent (and rather unexpected) applications of the theory of Gröbner bases to the study of the structure of solutions of linear systems of constant coefficients partial differential systems. Gröbner bases first appeared in Buchbergers's Ph.D. thesis [5] (see also [6] where the main results were first published), and their theory has provided the conceptual basis for the creation of several computational algebra packages which can be utilized for the solution of polynomial problems. The approach that I have successfully applied in a series of joint papers [1], [2], [3], [4], [10], [11] is made possible by the algebrization of analysis which began in the sixties [12], [16] and was then perfected by M. Sato and his collaborators in the seventies [20], [21]. In this introductory section, I will briefly recall the foundations of this algebraic approach to partial differential equations, while in section 2, I will show a few concrete and remarkable applications of Gröbner bases to specific systems of differential equations. I would like to point out that the way in which we have been using Gröbner bases is twofold: on one hand we have used some symbolic computation packages which are based on the theory of Gröbner bases; on the other hand (see Theorem 2), we have used the theory itself to generalize results which had been computed in special cases.

Introduction

One of the aims of Constructive Mathematics is to provide effective methods (algorithms) to compute objects whose existence is asserted by Classical Mathematics. Moreover, all proofs should be constructive, i.e., have an underlying effective content. E.g. the classical proof of the correctness of Buchberger algorithm, based on noetherianity, is non constructive : the closest consequence is that we know that the algorithm ends, but we don't know when.

In this paper we explain how the Buchberger algorithm can be used in order to give a constructive approach to the Hilbert basis theorem and more generally to the constructive content of ideal theory in polynomial rings over “discrete” fields.

Mines, Richman and Ruitenburg in 1988 ([5]) (following Richman [6] and Seidenberg [7]) attained this aim without using Buchberger algorithm and Gröbner bases, through a general theory of “coherent noetherian rings” with a constructive meaning of these words (see [5], chap. VIII, th. 1.5). Moreover, the results in [5] are more general than in our paper and the Seidenberg version gives a slightly different result. Here, we get the Richman version when dealing with a discrete field as coefficient ring (“discrete” means the equality is decidable in k).

Introduction

In the nineties, there is an increasing interest in combining symbolic and numerical methods. This can be seen at diverse instances. There are now international symposia supported by organizations from both sides, and the number of contributes displaying the symbolic - numerical interplay is increasing. Other examples are the facilities of using floating point arithmetics and simple numerical procedures in Computer Algebra Systems on the one hand and the (eventually partly) integration of Computer Algebra Systems into numerical software packages on the other hand. The most prominent example is here the migration of the Computer Algebra System AXIOM to NAG, the Numerical Algorithm Group.

Many interesting results have been obtained by combining symbolic and numerical methods, like in polynomial continuation the avoiding of solution paths diverging to infinity by means of concepts from toric ideals or like the numerical solving of systems of polynomial equations using resultants, see for instance Canny and Manocha (1993).

Introduction

It is well known that the term ordering strongly determines the complexity of the Gröbner basis computation. The choice of the term ordering usually depends on the type of problem we want to solve. Elimination orders such as lexicographic, which we need for polynomial system solving, are known to be slow term orders, that is, they lead to particularly long computations.

Remembrance

It was Autumn 1983, when the researchers on Gröbner could have been counted on the fingers of two hands. Michael and me were completing our algorithm to compute resolutions (Mora, Möller 1986a, 1986b) and I was invited in Naples to give an introductory tutorial on Gröbner bases.

I had plenty of free time and, since somebody had just quoted me the Eagon-Northcott formula expressing the resolution of the ideals generated by the majors of a matrix whose entries are independent variables (Eagon, Northcott 1962)), I decided to try to see whether our tools allowed me to tackle the 5 × 3 case.

I was really surprised when not only I got the resolution but I realized that it was sufficient to give a look to the solution to devise the complete formula (Th. 1.1) and that proving it required only to generalize the computation I did: it was the first time that I realized the amazing power of Buchberger's tool.

Introduction

Let R be the ring of all complex rational functions without poles in a given real interval. The work of U. Oberst and S. Fröhler ([7],[8],[14]) on systems of differential equations with time-varying coefficients raised several questions for modules over the ring R[D] of linear differential operators with coefficients in R.

There are a number of results ([2],[5],[6],[9],[11],[12],[13],[17],…) on Gröbner bases in rings of differential operators, but the coefficient rings are fields (of rational functions), rings of power series, or rings of polynomials over a field. In the latter case every differential operator is a K-linear combination of “terms” xi Dj, (i,j) ∈ Nn × Nn. Thus Gröbner bases are defined with respect to a term order on Nn × Nn, the coefficients are elements of a field and commute with the terms. This approach cannot be used for other coefficient rings (like R, for example).

The results of B. Buchberger ([3],[4]) on Gröbner bases in polynomial rings have been generalized by several authors (see for example [9]) to polynomial rings with coefficients in commutative rings. In analogy to this extension we present a basic theory of Gröbner bases for differential operators with coefficients in a commutative ring.

Preface

After the notion of Gröbner bases and an algorithm for constructing them was introduced by Buchberger [Bul, Bu2] algebraic geometers have used Gröbner bases as the main computational tool for many years, either to prove a theorem or to disprove a conjecture or just to experiment with examples in order to obtain a feeling about the structure of an algebraic variety. Nontrivial problems coming either from logic, mathematics or applications usually lead to nontrivial Gröbner basis computations, which is the reason why several improvements have been provided by many people and have been implemented in general purpose systems like Axiom, Maple, Mathematica, Reduce, etc., and systems specialized for use in algebraic geometry and commutative algebra like CoCoA, Macaulay and Singular.

The present paper starts with an introduction to some concepts of algebraic geometry which should be understood by people with (almost) no knowledge in this field.

In the second chapter we introduce standard bases (generalization of Gröbner bases to non–well–orderings), which are needed for applications to local algebraic geometry (singularity theory), and a method for computing syzygies and free resolutions.

The last chapter describes a new algorithm for computing the normalization of a reduced affine ring and gives an elementary introduction to singularity theory. Then we describe algorithms, using standard bases, to compute infinitesimal deformations and obstructions, which are basic for the deformation theory of isolated singularities.