An Introduction to Computational Algebraic Analysis

Fabrizio Colombo Dipartimento di Matematica Politecnico di Milano Via Bonardi, 9 20133 Milano, Italy fabcol@mate.polimi.it

Irene Sabadini Dipartimento di Matematica Politecnico di Milano Via Bonardi, 9 20133 Milano, Italy sabadini@mate.polimi.it

Daniele C. Struppa Department of Mathematical Sciences George Mason University Fairfax, VA 22030 USA dstruppa@gmu.edu

Abstract

In this paper we introduce the reader to the new field of computational algebraic analysis. We give some of the general ideas underlying this discipline, we show some of the key examples, we illustrate the applications which drive this area, and we conclude with some directions for further research.

1 Introduction

This paper provides an overview of the research of the authors (and some of their coworkers) in the last few years. While our work began in the attempt of studying analytic properties of functions of several quaternionic variables, we quickly realized that our ideas and methods could be expanded to help make effective the fundamental work of the pioneers in algebraic analysis. In the last couple of years, we have realized that a new field, which we called computational algebraic analysis, was being born, and we are currently in the process of concluding a book [14] on this topic. This paper may be considered as an introduction to the work described in the book.

After a first section in which we discuss the basic notions of algebraic analysis, we show, in section three, how a sizeable part of the theory of several complex variables can be interpreted in terms of algebraic analysis. We then devote section four to describing the challenges which the study of more complicated situations offer, and we discuss the nature of computations which need to be addressed.

The systems of differential equations which we study with the tools of computational algebraic analysis have a particular interest for theoretical physicists, and we dedicate section five to a discussion of the physical meaning and implications of our work. Finally, we conclude with a section in which we discuss some of our most recent results, and how they pave the way for a new set of problems, and challenges.

This paper is an outgrowth of the seminar which the third author gave at the University of Milano on November 2002. The third author wishes to thank the Department of Mathematics, for the invitation, and George Mason University for the very generous leave of absence which allowed him to travel to deliver this talk.

2 What is Algebraic Analysis?

The term ”Algebraic Analysis” was probably first used by Euler in his fundamental works on differential equations. Those works, which lay the foundations for the modern theory of ordinary differential equations, contain, in particular, a result which is the prototype of algebraic analysis, and which we now teach to our students in their first course on differential equations. The result, which is often referred to as Euler’s Fundamental Principle, is a result which has inspired much of what we will discuss in this paper and it may be worth stating it, after introducing some notations.

Let P (z) be a complex polynomial in one variable. The set of solutions to the algebraic equation P (z) = 0 can be described as a finite set of pairs (z_{k}, m_{k}), where z_{k} are the roots of P and m_{k} denote their multiplicities. The index k will take values from 1 to some integer K, but the sum of all m_{k}’s will always equal the degree of the polynomial P (this is, of course, the Fundamental Theorem of Algebra). This same polynomial can be naturally considered as the symbol of the differential operator obtained by formally replacing z by the derivative d/dx. In the study of linear ordinary differential equations, one is interested in studying, for example, the set of all smooth functions f on ℝ such that

P (d/dx)f = 0.

Euler’s theorem can then be stated as follows:

Theorem 2.1. Any smooth solution to the differential equation

P (d/dx)f = 0

can be represented as

f(x) = ∑_{k} c_{k}(x)e^{z_{k}x}

where the c_{k}(x) are arbitrary polynomials in x of degree strictly less than m_{k}.

It is clear, then, why we said that Euler’s Fundamental Principle is the first result of algebraic analysis. Indeed, the theorem states that in order for us to fully describe the kernel of the differential operator P (d/dx), we only need to fully understand the zeroes of the polynomial P, an inherently algebraic object.

The theorem, of course, says much more, and in particular establishes that exponentials and polynomials are the only important functions in the study of linear, constant coefficients, ordinary differential equations, a fact which Ehrenpreis will extend to the case of partial differential equations, but which is clearly not true when variable coefficients are considered.

The term ”Algebraic Analysis”, however, has had an important revival with the Japanese school of Kyoto, led by Sato, Kawai, Kashiwara and their coworkers. These mathematicians developed, beginning in the early sixties, an extremely deep theory of differential equations, fully based on an algebraic treatment. But, as one can read in the introduction to the famous [26], there is more (in their eyes, at least) to algebraic analysis than the use of algebra to solve analytic problems. In fact, according to Kawai, Kashiwara, and Kimura, algebraic analysis is “that analysis which holds onto substance and survives the shifts of fashion in the field of analysis, as Euler’s mathematics, for example, has done”.

This little historical discussion, however, has done little to answer the question in the title of this section. To provide such an answer, one may want to make reference to another important historical event, namely the birth of modern algebraic geometry, with the fundamental works of Hilbert in the early nineties (we are talking here 1890). In fact, between 1890 and 1893, Hilbert proved a series of fundamental results, see [25], which allowed the creation of algebraic geometry as we know it now. The results which we have in mind are the Basis Theorem (essentially the noetherianity of the ring of complex polynomials), the Nullstellensatz (or Theorem of the Zeroes, in which one shows the link between the geometry of varieties and the algebra of ideals in polynomial rings), and finally the Syzygy Theorem, which is the fundamental result for this paper, and which we will describe in detail later on.

It was thanks to those works, that it became clear that the study of inherently geometrical objects such as curves and surfaces (and whatever else the imagination of geometers could envision) could be enhanced (in fact, replaced) by the study of inherently algebraic objects such as ideals and modules in rings of polynomials. So, instead of looking at the points where a given polynomial vanishes (its algebraic set), one could study the ideal generated by that same polynomial. This alternative point of view has the advantage of invariance (different polynomials, in fact all the polynomials in the ideal, can give the same geometrical object), and also provides the mathematician with all the toolkit of the algebraist. The validity of the approach has been demonstrated beyond any doubt by the incredible success of algebraic geometry in the last one hundred years, and by its continuous promises of growth and development.

What happened in analysis towards the late fifties (maybe beginning with a beautiful, though little known, paper of Ehrenpreis published in 1956, [17]), was a movement guided by the same principles. In analysis, one of the important objects of study is the kernel of a differential operator. But different operators may originate the same kernel, and so an invariant description seems of interest here as well. So, the idea behind the work of the early mathematicians in algebraic analysis (without pretense of completeness, we would like to mention Malgrange, Ehrenpreis, Palamodov, Hörmander), consisted in replacing systems of differential equations with suitable algebraic objects.

To describe the necessary approach (and finally understand what we mean by algebraic analysis), we introduce a few notations. Let R denote the ring of complex polynomials in n variables, i.e. R = ℂ[z_{1}, …, z_{n}]. As in the case of Euler’s Fundamental Principle, we can regard R as the ring of symbols of linear constant coefficients partial differential operators. To realize this, we formally replace z = (z_{1}, …, z_{n}) with D = (∂/∂x_{1}, …, ∂/∂x_{n}). The formal substitution of derivatives with polynomials and viceversa is allowed by the Fourier transform only in spaces of functions satisfying suitable conditions. Among those are the so called Localizable Analytically Uniform spaces (LAU spaces), and those are the spaces for which Ehrenpreis’ Fundamental Principle holds.

Among the linear topological spaces which admit a LAU structure we recall

• The space of polynomials in n variables with complex coefficients.

• The space O(Ω) of holomorphic functions on an open convex set Ω ⊆ ℂ^{n}.

• The space of entire functions of exponential type.

• The space C^{∞}(Ω), where Ω is an open convex set in ℝ^{n}.

• The space of distributions D′(Ω), where Ω ⊆ ℝ^{n}.

• The space D′_{ω}(Ω) of Beurling ultra-distributions.

One may now consider an r_{1}×r_{0} matrix P = [P_{ij}] of elements in R. Such a matrix is obviously the symbol of a matrix of differential operators P (D), and if S is a space of generalized functions, one sees that P (D) defines a natural map

P (D) : S^{r_{0}} → S^{r_{1}},

whose kernel is the object in which analysts are interested.

As it had already happened in geometry, however, it soon became clear among analysts as well that the best way to treat situations of this sort was to introduce the notion of sheaf, so that from now on, we will not consider simply spaces, but rather sheaves of generalized functions, to which the entire algebraic theory will be applied. Partial differential operators have the important property of acting on sheaves as sheaf-homomorphisms, and so if S is such a sheaf, we now have that P (D) is a sheaf homomorphism whose kernel (again a sheaf) we customarily denote by S^{P}.

In the sequel we will consider several different sheaves of generalized functions. While the theory we will describe works for any such sheaf, the reader may want to think of some fundamental examples such as the sheaf A of real analytic functions, the sheaf B of hyperfunctions, the sheaf C of microfunctions (this is a sheaf defined on the cotangent bundle to the variety on which one normally studies differential equations, and its treatment requires some additional comments which will not find space in this paper), the sheaf D′ of Schwartz distributions, the sheaf E of infinitely differentiable functions, and finally the sheaf O of holomorphic functions.

The first important statement in algebraic analysis is the fact that the algebraic object we must be interested in is the R-module

M = R^{r_{0}}/P^{t}R^{r_{1}}

where P^{t} denotes the transpose of P. The argument behind the centrality of this module is given by the following fundamental theorem (in a way the theoretical basis for all of algebraic analysis):

Theorem 2.2. Let S be a sheaf of generalized functions. Then there is a sheaf isomorphism

S^{P} ≅ Hom(M, S).

Proof. We only give a sketch of the proof, and leave it to the reader the task of completing it in the general case. We will assume r_{0} = 1, so that the solutions to P (D)f = 0 are actually functions and not vectors of functions. Also, we will consider the case in which we look at global solutions (rather than germs of solutions as working with sheaves would require). Let then f be a solution in S of the equation P (D)f = 0. We can construct the morphism σ_{f} from R to S by mapping the identity 1 to the function f. It is easily seen that because P (D)f = 0, then σ_{f} is actually an R-homomorphism defined on M. One can then show that the correspondence f → σ_{f} is the isomorphism one needs to define. □

Remark 2.3. This result clearly indicates why M is the central object in algebraic analysis, as it shows that the study of solutions of a general system of (linear, constant coefficients) partial differential equations can be reduced to the purely algebraic task of classifying all the morphisms from M to the space in which we want to find the solutions.

Remark 2.4. So far, and in what follows, we have always operated with the ring of polynomials, which in terms of analysis corresponds to working on linear constant coefficients partial differential operators of finite order. However, we want to point out that the same theorem holds if we replace R with the Weyl algebra (in which case we are working with linear differential operators with polynomial coefficients), with the sheaf D of differential operators with real analytic coefficients, or even with the sheaf of differential operators with real analytic coefficients and infinite order (this is the full context for algebraic analysis in the sense of the Japanese school).

Now that we have established that the module M is the algebraic object which incorporates all the interesting information on the solutions of the system of differential equations P (D)f = 0, we need to understand how to study such a module. Here, as we mentioned before, is where the traditional tools from commutative algebra come into the picture, in a very similar way to what had happened for algebraic geometry.

The first important step, indeed, is to realize that one can always write a finite resolution of the module M, according to what is usually called Hilbert’s syzygy theorem. This fundamental result can be stated as follows.

Theorem 2.5. There exists an integer m ≤ n and a finite exact resolution of the module M with free modules as follows:

0 ⟶ R^{r_{m}} ⟶^{P_{m−1}^{t}} R^{r_{m−1}} ⟶ … ⟶^{P_{1}^{t}} R^{r_{1}} ⟶^{P^{t}} R^{r_{0}} ⟶ M ⟶ 0.  (1)

The maps which appear in this resolutions are called the syzygies of M, and they can be constructed in several different ways, so that the importance of the result is not the existence of a resolution, but the fact that one can find a finite resolution, as well as the fact that we have a natural bound on its length. It is important to note, however, that such a resolution is not unique.

One can then dualize such a resolution through the use of the Hom functor (essentially we take the duals of the spaces involved, we take the transpose of the matrices representing the operators, and we reverse the arrows) to obtain:

0 ⟶ R^{r_{0}} ⟶^{P} R^{r_{1}} ⟶^{P_{1}} … ⟶ R^{r_{m−1}} ⟶^{P_{m−1}} R^{r_{m}} ⟶ 0  (2)

Since (for those who speak category language) the Hom functor is not exact, the complex we have just obtained is not necessarily exact, so that one can consider its cohomology (the measure of how inexact the complex is) by taking the quotients of kernels and images. The quotient groups one obtains are actually R-modules:

Definition 2.6. The Ext–modules of M are defined as:

Ext^{j}(M, R) = H^{j}(M, R) = \frac{Ker(P_{j})}{Im(P_{j−1})}.

Remark 2.7. It is a fundamental point of the entire theory the fact that while the syzygies of Hilbert’s theorem are not uniquely defined, the Ext–modules are uniquely determined by M and R, and thus are the invariant algebraic objects we are looking for.

Remark 2.8. Sequence (2) is very important to analysts because it can be given an immediate analytical reading. If S is a sheaf of generalized functions, and if we maintain the same notations used up to now, we have that for every open set U, the following sequence is exact [28]:

0 ⟶ S^{P} (U) ⟶ S^{r_{0}}(U) ⟶ S^{r_{1}}(U) ⟶ … ⟶ S^{r_{m−1}}(U) ⟶ S^{r_{m}}(U) ⟶ 0

Much of the traditional algebraic analysis is based on the study of these modules and on the understanding that their properties tell us fundamental things about the system P (D)f = 0. The basic results in this direction are due to mathematicians such as Ehrenpreis, Malgrange, and Palamodov, and probably the best exposition of all the relevant properties can be found in [32] or in the more recent work of Komatsu [28]. We will come back to some of these results in the next section, but before we proceed, we want to point out the main ’weakness’ of the theory (if we can use such a strong word) as it stood in the seventies. Even though one could prove extremely general results through the use of the Ext-modules, the real difficulty resided in the fact that it was virtually impossible to compute such modules in any case of real interest. We recall the pleasant surprise that our first work on the subject provoked in Palamodov when we showed how we could finally easily compute all the Ext-modules for a rather complicated system which arises in significant contexts (the system is the Cauchy-Fueter system of which we will speak in a while).

So, we want to emphasize how the beauty of the theory was, until recently, somewhat undermined by the difficulty to use tools which (though powerful in abstract) were resistant to simple and direct computations.

Finally, to conclude this short section on algebraic analysis, we wish to discuss the culminating and most important result of this algebraic theory, independently obtained and proved by Ehrenpreis and Palamodov, the so called Fundamental Principle of Ehrenpreis-Palamodov. This principle allows to write in an explicit way the general solution of a homogeneous system of constant coefficients differential equations. It is a very far reaching generalization of Euler’s Fundamental Principle (Theorem 2.1).

In order to fully state the Ehrenpreis–Palamodov’s principle, we need to introduce the notion of multiplicity variety associated to a system. Such a variety is described in detail in [19], [32], but it can quickly be explained as follows.

Theorem 2.9. Let P be a r_{1} × r_{0} matrix of polynomials. There exists a vector

B⃗ = {B_{1}, …, B_{s}} = {(V_{1}, δ_{1}); (V_{2}, δ_{2}); …; (V_{s}, δ_{s})}

of algebraic varieties V_{j} and s_{j} × r_{0} matrices δ_{j} of polynomial coefficients differential operators for j = 1, …, s, such that F⃗ ∈ ⟨P^{t}R^{r_{1}}⟩ if and only if δ_{j}F_{|V_{j}} = 0.

Remark 2.10. Note that the previous theorem is a very powerful generalization of the Hilbert Nullstellensatz.

Remark 2.11. The vector B⃗ which appears in Theorem 2.9 is known as the multiplicity variety associated to the matrix P (see [19]). Its construction can be carried out if one knows the associated set Ass(M) of the R–module M, i.e. the set of all prime ideals in R that are annihilators of elements in M. For each ideal ℘_{j} ∈ Ass(M) there exists a differential operator δ_{j} : R^{r_{0}} → (R/℘_{j})^{s_{j}}, known as noetherian operator for M that can be obtained as the composition of the maps

∂_{j} : R^{r_{0}} → R^{s_{j}}, π : R^{s_{j}} → (R/℘_{j})^{s_{j}}

where π is the natural projection and the matrix ∂_{i} has entries that are polynomials in the variables x_{i} and their derivatives. The varieties V_{j} are then defined by the common roots of the polynomials in ℘_{j}. The variety V = ∪V_{j} is known as the characteristic variety of the R-module M, see [32], Chapter IV, and it coincides with the set of points for which rank(P) < r_{0}.

We are now in position to state the Ehrenpreis-Palamodov Fundamental Principle.

Theorem 2.12. Let W be a LAU-space on an open convex set Ω. Let P⃗ (D) : W^{r_{0}} → W^{r_{1}} be a linear differential operator with constant coefficients, let B⃗ be its multiplicity variety and let ∂_{j} be such that ∂_{j} · π = δ_{j}. Then an element F⃗ ∈ W^{r_{0}} is a solution of

P⃗ (D)F⃗ = 0

if and only if

F⃗ (x) = ∑_{j=1}^{s} ∫_{V_{j}} exp(iz · x) ∂_{j}^{t} μ_{j}(z),

where exp(iz · x) is a column vector with r_{0} components, ∂_{j}^{t} is the transpose of ∂_{j} and μ_{j}(z) is a ℂ^{s_{j}}-valued density with support in V_{j} such that setting p_{K}(ξ) = max_{x∈K} Re(ξ · x)

∑_{j=1}^{s} ∫_{V_{j}} exp(p_{K}(ξ))|μ_{j}| < ∞

for any compact set K ⊂ Ω.

Remark 2.13. The representation above must be correctly interpreted. So, for example, if we are working with distributions or hyperfunctions, the integrals must be interpreted in the functional sense.

3 Several Complex Variables and Algebraic Analysis

In this section we show how the general theory of several complex variables can be fully reconstructed as a (more or less) immediate application of the algebraic analysis ideas expressed in the previous section. The reader should understand, however, that the historical developments are very different from the way in which we present the results in this section. Much of the theory in several complex variables was developed through hard and careful analytic work. It was only in the sixties that it became clear how much of the theory could really be reinterpreted in these terms.

To begin with, we need to say a few words about the construction of the syzygies for a particular system of differential equations or, equivalently, for a given matrix. If, as in section two, we look at a generic rectangular matrix, then the construction of the syzygies is really a very complicated task. However, if one limits oneself to a matrix in which r_{0} = 1, and if one makes a few additional algebraic hypotheses, then it turns out that the task becomes much more manageable.

Example 3.1. Let us then take a quick look at what would happen if the matrix P were in fact a vector of, say, four polynomials, and let us assume (algebraic condition) that these polynomials have no common factors (in fact, one might ask for less than this, but we are not looking for optimal statements at this stage).

We then have than M = R/I(P_{1}, P_{2}, P_{3}, P_{4}), where I is the ideal generated in R by the four polynomials P_{1}, ..., P_{4}. The Hilbert’s resolution then begins as follows:

R^{4} →^{P^{t}} R → M → 0

where the map from R^{4} to R is the map obtained by multiplication with P^{t}. The next step in building Hilbert’s resolution is to describe the kernel of such map in R^{4}. It is obvious that there are a few 4-vectors which are necessarily in the kernel. For example, the vector (−P_{2}, P_{1}, 0, 0) is in the kernel, and so is the vector (0, −P_{3}, P_{2}, 0), etc. In fact, one can easily construct 6 = \binom{4}{2} such elements in the kernel for any choice of a pair (P_{i}, P_{j}). It can be shown that if the polynomials in P form a regular sequence (in particular if they have no common factors), then the kernel is generated by these six vectors, and we have therefore extended our resolution to one more step as follows:

R^{6} →^{P_{1}^{t}} R^{4} →^{P^{t}} R → M → 0

where the first map is now given by the 6 × 4 matrix

P_{1} = [\begin{matrix}−P_{2} & P_{1} & 0 & 0 \\ −P_{3} & 0 & P_{1} & 0 \\ −P_{4} & 0 & 0 & P_{1} \\ 0 & −P_{3} & P_{2} & 0 \\ 0 & −P_{4} & 0 & P_{2} \\ 0 & 0 & −P_{4} & P_{3} \end{matrix}]

One now needs to construct the kernel of such operator. This kernel lies in R^{6} so it can be constructed by taking a generic vector in R^{6}, and setting equal to zero its product with the matrix above. Using repeatedly the property of regularity of the sequence P (again, lack of common factors is a sufficient condition for this), one can verify (though it may take a few minutes) that the kernel of this map is given by the 4 × 6 matrix

P_{2} = [\begin{matrix}P_{3} & −P_{2} & 0 & P_{1} & 0 & 0 \\ P_{4} & 0 & −P_{2} & 0 & P_{1} & 0 \\ 0 & P_{4} & −P_{3} & 0 & 0 & P_{1} \\ 0 & 0 & 0 & P_{4} & −P_{3} & P_{2} \end{matrix}]

This leads us to one more step in the resolution, namely

R^{4} →^{P_{2}^{t}} R^{6} →^{P_{1}^{t}} R^{4} →^{P^{t}} R → M → 0.

We can finally look for the kernel in R^{4} of this last map we have constructed. Arguing as above, one obtains quickly that such a kernel is given by the 1 × 4 matrix

P_{3} = [−P_{4}, P_{3}, −P_{2}, P_{1}].

This new operator, obviously, has trivial kernel and so the resolution in Hilbert’s theorem is given by:

0 → R →^{P_{3}^{t}} R^{4} →^{P_{2}^{t}} R^{6} →^{P_{1}^{t}} R^{4} →^{P^{t}} R → M → 0.

The resolution we have constructed is well known in algebra, and goes under the name of Koszul complex.

Several comments are in order. First we note that the same exact process would work for any number of polynomials, so that the case n = 4 is really quite general. Second we note an interesting feature in the Koszul complex, in fact several interesting features. The different maps are all of degree one in the polynomials P_{j}; the order of the matrices provides for a symmetrical complex, whose Betti numbers (the exponents of R) are given by the binomial coefficients; finally, the last matrix of the complex is (up to signs and ordering) exactly the transpose of the matrix we started with. In particular, if we consider the associated differential operators, we have that the first and the last operators are the same, in terms of their solution spaces.

Something more (and more refined) could be said about these complexes. One can show, by using this resolution, that all the Ext-modules of M vanish, with the exception of the 4-th one. More in general, if we were to have n polynomials, one could verify that all the Ext vanish, with the exception of the n-th one.

To bring this discussion to the level of several complex variables, one simply notices that a (real) differentiable function f on ℂ^{n} is said to be holomorphic if it satisfies the Cauchy-Riemann system of differential equations given by

\{ \begin{matrix}\frac{∂}{∂z̄_{1}} f = 0 \\ … \\ \frac{∂}{∂z̄_{n}} f = 0 \end{matrix}

where

z_{ℓ} = x_{ℓ} + iy_{ℓ}  and  \frac{∂}{∂z̄_{ℓ}} = \frac{∂}{∂x̄_{ℓ}} + i\frac{∂}{∂ȳ_{ℓ}} ,  for ℓ = 1, …, n.

It is then very easy to see that the study of holomorphic functions is nothing more than the study of the solutions of the Cauchy-Riemann system, and as such we need to consider the associated Koszul complex. The Cauchy-Riemann system clearly satisfies the conditions which we have indicated during the construction of the Koszul complex; namely, we have a system with a single unknown function f, and we have that the n polynomials have no common factors. In fact, in this case we have one additional useful property, i.e. the fact that all the polynomials are of degree 1 and that they utilize different variables. These facts, while quite useful, are not central to our construction.

It is easy to construct the Koszul complex associated to this complex. According to what we have shown in the preceding part of this section, we have the following exact sequence

0 ⟶ O(U) ⟶ C^{∞}(U) ⟶^{[ \frac{∂}{∂z̄_{1}} ,..., \frac{∂}{∂z̄n}]} (C^{∞})^{\binom{n}{1}}(U) ⟶^{P_{1}(D)} … ⟶^{P_{n−1}(D)} (C^{∞})^{\binom{n}{n}}(U) ⟶ 0.

Remark 3.2. There is a very interesting observation to make at this point, which we will find of some use later on. In complex analysis, one can develop a theory of holomorphic differential forms, which gives rise to a sequence known as the Dolbeault sequence. If one considers the differential forms of type (p, q) i.e. elements of the form

ω = ∑ ϕ_{i_{1}...i_{p}j_{1}...j_{q}}dz_{i_{1}} ∧ … ∧ dz_{i_{p}} ∧ dz̄_{j_{1}} ∧ … ∧ dz̄_{j_{q}},

and if one denotes the space of (p, q)–forms with C^{∞} coefficients on an open set U ⊆ ℂ^{n} by E^{p,q}(U), one has the following Dolbeault sequence, in which the maps are the usual differentials

¯∂ω = ∑_{i} \frac{∂ϕ_{j_{1}...j_{q}}}{∂z̄_{i}} dz̄_{i} ∧ dz_{i_{1}} ∧ … ∧ dz_{i_{p}} ∧ dz̄_{j_{1}} ∧ … ∧ dz̄_{j_{q}}

see [29] for the details:

0 ⟶ O(U) ⟶ E^{0,0}(U) ⟶^{¯∂} E^{0,1}(U) ⟶^{¯∂} … ⟶^{¯∂} E^{0,n}(U) ⟶ 0.

The rather striking fact which one can observe is that the two complexes we have just written are just the translation one of the other. In fact, by the way in which the maps are constructed in the Koszul complex, one can see that they act exactly as the differentials defined in the Dolbeault sequence. We will see later on how this behavior can inspire additional ideas in a variety of situations.

Remark 3.3. Another important fact is that the map P_{r+1}(D) in the resolution of an operator P (D) gives the compatibility conditions on the datum g of the non homogeneous system

P_{r}(D)f = g that assure the solvability of the system. In particular when P (D) = [\frac{∂}{∂z̄_{1}} , …, \frac{∂}{∂z̄_{n}}],

the matrix P_{1}(D) constructed as above, gives the well known compatibility conditions

\frac{∂}{∂z̄_{j}} g_{i} − \frac{∂}{∂z̄_{i}} g_{j} = 0

on the datum g = [g_{1}, …, g_{n}] of the system P (D)f = g.

The question we can now reasonably ask, is what kind of analysis we can deduce from the algebra hidden in these sequences. The various algebraic properties one can see have a (more or less) immediate consequence.

First, it was proved by Ehrenpreis in a seminal paper in 1961 (see [18], probably the first major result in modern algebraic analysis) that the vanishing of Ext^{1}(M, R) is equivalent to the fact that the generalized functions solutions of the associated system outside of a compact set, automatically satisfy the same system within that same compact (in other words, solutions of systems with vanishing Ext^{1}(M, R) cannot have compact singularities). When one considers the case of the Cauchy-Riemann system, this entails that if a function is holomorphic (in several variables) outside of a compact set, it is automatically holomorphic inside that same compact set. This is a very classical and fundamental result which was proved by Hartogs in 1906 [24], and which is by many considered the first important result in the theory of several complex variables. What Ehrenpreis proved, therefore, is that an exquisitely analytical result such as Hartogs compact singularities theorem, is in fact a purely algebraic result, which holds not just for holomorphic functions, but rather for a wide variety of spaces of functions.

The vanishing of all the other Ext modules is also significant as it allows for the removability of compact singularities of various cohomology classes (see e.g. [32] for a deeper discussion on this topic). However, this vanishing entails several more significant results. The first deals with the vanishing of the cohomology of open sets with coefficients in the sheaf of solutions of the system. In the case of the Cauchy-Riemann system in several complex variables, one can show that the vanishing of the Ext modules allows the proof of the celebrated Malgrange theorem which states that, for any open set U in ℂ^{n}, and for every j ≥ n, it is

H^{j}(U, O) = 0.

This result, clearly, generalizes the Mittag-Leffler theorem which, in one variable, states the vanishing of H^{1}(U, O). Interestingly enough, it was exactly Hartogs’ theorem which showed that the appropriate generalization for Mittag-Leffler theorem could not be the vanishing of H^{1} in n variables. It took Malgrange’ s result, however, to demonstrate that the appropriate way to phrase Mittag-Leffler is not in terms of the index 1, but rather in terms of any index j greater than or equal to the dimension n of the ambient space.

One more important result has to do not just with the vanishing of the Ext–modules, but also with the symmetry of the system. Under these conditions, in fact, one can prove an important duality theorem. Let S be a sheaf of generalized functions, and let P = (P_{1}, …, P_{n}) be a vector of polynomials as in the construction we have given above. Finally, let S^{P} denote the sheaf of S-solutions to the system P (D)f = 0. Then under the conditions discussed above, one has the following isomorphism for every compact set K:

(S^{P} (K))′ ≅ H_{K}^{n}(ℂ^{n}; S^{P}).

In this isomorphism, the right hand side denotes the n–th cohomology group with support; the reader is referred to [28] for details. In more general cases (see for example [27]) one has an isomorphism in which the cohomology classes are taken in a different sheaf, namely the so called dualizing sheaf for S; this sheaf arises in a natural way from the last operator which appears in the resolution. However, the symmetry of the Koszul complex in this case allows us to have the same sheaf on both sides of the isomorphism, for a more satisfying statement.

In the particular case in which one deals with the Cauchy-Riemann system, the isomorphism was first noted by Sato during his construction of hyperfunction theory in the late fifties. From this perspective, the space (S^{P} (K))′ is nothing but the space (O(K))′, i.e. the space of analytic functionals carried by K, which is isomorphic, through this result, to the space of hyperfunctions supported in K. When n = 1 this result reduces to the well known Köthe duality theorem for compact subsets K of ℂ:

(O(K))′ ≅ O(ℂ \ K)/O(ℂ).

It is obvious, once again, that this result cannot be extended as it appears to several complex variables (again a consequence of Hartogs’ theorem on the removability of compact singularities). However, the cohomological version we have stated is its perfect n-dimensional analogue.

4 Moving beyond the Cauchy-Riemann system

In this section we show that what can be easily done in the case of the Cauchy-Riemann system, can also be attempted for more complex systems. However, as we shall see, one may encounter significant more difficulties related both to the specific form of the system, and to the dimensions which are involved. It is during this process that we have developed the first tools of computational algebraic analysis which we discuss in this paper.

The first system we tried to study beyond the Cauchy-Riemann system is a very simple generalization which has interesting historical origins, and whose study ends up being much harder than one might expect at first sight.

Back in the forties, the Swiss mathematician Fueter decided to investigate the properties of functions defined on the space of quaternions ℍ, [22].

Quaternions, defined as elements of the form q = x_{0} + ix_{1} + jx_{2} + kx_{3} with i, j, k imaginary units (i,e, i^{2} = j^{2} = k^{2} = −1 and satisfying the well known relationships ij = k, jk = i, ki = j, ji = −k, kj = −i, ik = −j), had been introduced by Hamilton in the first half of 19th century, and had become rapidly an important object of study because they offered the first significant example of a non-commutative structure in mathematics. Quaternions obviously have a sort of similarity with complex numbers, and so it was only natural that mathematicians attempted to study, on ℍ, a notion similar to the notion of holomorphicity for functions of a complex variable. As it is well known, one can think of holomorphic functions in three different (though ultimately equivalent) ways. A function on ℂ is holomorphic if it admits complex derivative; a function on ℂ is holomorphic if it admits a power series representation; finally a function is holomorphic if it satisfies the Cauchy-Riemann system. It is relatively easy to show that a generalization of holomorphicity to ℍ based on the notion of quaternionic derivative would fail to be interesting, as the only functions which admit quaternionic derivative are the linear functions. Similarly, a generalization based on the notion of power series (because of non commutativity this means that the monomials of degree n are of the form a_{0}qa_{1}q … a_{n}q) fails to be interesting because reduces to the theory of analytic functions in the four real variables x_{0}, …, x_{3}. So, it became quickly clear that in order to define holomorphicity in ℍ (actually, the preferred term is regularity), one had to find an appropriate system of differential equations. Such a system was described first by Fueter, and is known with the name of Cauchy-Fueter system.

We define the operator \frac{∂}{∂q̄} as follows:

\frac{∂}{∂q̄} = \frac{∂}{∂x_{0}} + i\frac{∂}{∂x_{1}} + j \frac{∂}{∂x_{2}} + k \frac{∂}{∂x_{3}} .

For a real differentiable function f : U ⊆ ℍ → ℍ, we say that f is regular in the open set U if it satisfies the Cauchy-Fueter system, i.e. if

\frac{∂}{∂q̄}f = 0.

Both the function f and the operator ∂/∂q̄ can be written in real coordinates, if one considers f not so much as a function from ℍ to ℍ, but rather as a function from ℝ^{4} to ℝ^{4}. From this point of view, the Cauchy-Fueter system becomes a 4 × 4 matrix system acting on a 4-vector of functions. Thus, we see that we now have a system of concrete interest, which needs to employ the general algebraic analysis theory in order to be studied.

Remark 4.1. Our interest for complexes of systems of hypercomplex operators goes back to our first paper on the topic [1] when we began studying non-homogeneous Cauchy–Fueter systems on quaternions (but see [7] for the first announcement of the results). The lack of commutativity typical of the quaternionic situation implied that the compatibility conditions necessary and sufficient for the solutions of such systems were unusual and somewhat unexpected.

Remark 4.2. It is interesting to recall that one of the reasons why Fueter was interested in this system is the fact that, originally, he thought one could use functions of one quaternionic variable to better understand the behavior of functions of two (and maybe more) complex variables. In particular (and this is almost ironic), Fueter was interested in finding a “better” proof of the Hartogs’ theorem, see [21]. The original proof given by Hartogs in 1906, was a typical example of several complex variables in the early century, when little more than simple Cauchy integrals were available. In fact, the only existing Cauchy formula for several complex variables was the formula for the polydisc, which is hardly more than an iteration of the one-dimensional Cauchy formula. It was by using formulas of this nature, and lots of ingenuity, that Hartogs was able to prove his groundbreaking result.

Fueter, however, believed that the removability of compact singularities had to be the expression of some deeper property of several complex variables, possibly linked to some new integral formulas. He also believed (see [41] for some historical analysis of these ideas) that the study of quaternions could shed some light, in view of the fact that every quaternion q can be thought of as two complex numbers, through the natural relation

q = x_{0} + ix_{1} + jx_{2} + kx_{3} = (x_{0} + jx_{2}) + i(x_{1} + jx_{3}).

Fueter developed a theory for regular functions, and showed that for such functions it is possible to build a Cauchy kernel (or a Cauchy-Fueter kernel as it is known now), and a corresponding Cauchy (actually Cauchy-Fueter) integral formula. Using this formula Fueter proved that the Hartogs’ theorem could be given a different and more intrinsic proof. Even more interesting is the fact that the ideas of Fueter were extremely influential on two of the best mathematicians of the forties, who explicitly used his papers to develop (independently and almost simultaneously) some new and original integral formulas for several complex variables. We now know those formulas from the names of those mathematicians, and we call them the Martinelli-Bochner formulas.

Remark 4.3. For those with an interest in the development of this theory, it is worth pointing out that the Cauchy-Fueter operator had already appeared (though in slightly different form) in the work of two Rumanian mathematicians, namely Moisil and Theodorescu (see [31]). The system they studied has been the object of a modern treatment in [30] and, from an algebraic analysis point of view, in [35].

When we originally looked at the literature on the properties for solutions of the Cauchy-Fueter system, we realized that the theory was fairly complete for the case of one quaternionic variable (see e.g. [16]), but on the other hand very little was known and understood for the case of several quaternionic variables (with the exception of the interesting but incomplete work of Pertici [33]). We decided to try to implement the techniques of algebraic analysis by beginning with the simple case of two quaternionic variables (see [7] and [1]).

It became immediately clear that the analysis of the 8 × 4 real system which corresponds to Cauchy-Fueter in two variables was a really difficult task from a computational point of view. To begin with, no such thing as the Koszul complex could be used. While a generalized Koszul complex is available (see the original works of Buchsbaum [10] for its description, or our own work [40] for its utilization in analysis), it was soon apparent that its use would not have led to easily implementable results. For one thing the computations were really lengthy (not a surprise, since a well known paper of Bayer and Stillman [5] indicates that the complexity of the computation of syzygies is doubly exponential), but also the syzygies which one would obtain were of degree at least three, which somehow contrasted with our intuition.

In fact, a simple analytical argument could be used to show that at least 8 real syzygies existed, and that they had a very simple quaternionic interpretation. Consider the operators ∂/∂q̄_{1}, ∂/∂q̄_{2} and their right inverses T_{1}, T_{2}. If f is a solution of the system

\{ \begin{matrix}\frac{∂}{∂q̄_{1}} f = g_{1} \\ \frac{∂}{∂q̄_{2}} f = g_{2} \end{matrix}

then there are functions h_{1}, and h_{2} regular in q_{1} and q_{2} respectively for which

f = T_{1}g_{1} + h_{1} = T_{2}g_{2} + h_{2}.

Therefore,

T_{1}g_{1} − T_{2}g_{2} = h_{2} − h_{1}

is the sum of a function regular in q_{1} and a function regular in q_{2}. Since the Laplacian operator Δ_{i} in q_{i} is a real operator that factors as

Δ_{i} = \frac{∂}{∂q̄_{i}} \frac{∂}{∂q_{i}} = \frac{∂}{∂q_{i}} \frac{∂}{∂q̄_{i}}

for i = 1, 2, we have that h_{1} − h_{2} is in the annihilator of the module generated by Δ_{1} \frac{∂}{∂q̄_{2}} and Δ_{2} \frac{∂}{∂q̄_{1}} . This argument shows that the matrix of syzygies contains at least the following 2 × 2 matrix

[ \begin{matrix}\frac{∂^{2}}{∂q̄_{2}∂q_{1}} & −\frac{∂^{2}}{∂q̄_{1}∂q_{1}} \\ \frac{∂^{2}}{∂q̄_{2}∂q_{2}} & −\frac{∂^{2}}{∂q̄_{1}∂q_{2}} \end{matrix} ] .

However, there is no easy analytical way to show that these are the only syzygies which will be needed. Indeed, we would like to pose this as an interesting research problem

Research Problem 1. How can we prove, by using only analytical arguments, that the syzygies which one obtains algebraically (both in the case of two and in the case of several variables) are in fact all the syzygies one needs.

As for us, we resorted to our first venture in what we call ’experimental mathematics’, by utilizing CoCoA^{1} to compute the resolution for the Cauchy-Fueter system in two, three, and four variables. The results we obtained for two variables were exciting, though somewhat misleading. The resolution we obtained was, in real coordinates,

0 ⟶ R^{4}(−4) ⟶ R^{8}(−3) ⟶ R^{8}(−1) ⟶ R^{4} ⟶ M_{2} ⟶ 0

where we denote by M_{n} the R-module associate to the Cauchy-Fueter system in n quaternionic variables, the ’exponents’ of the ring R indicate the number of syzygies at each stage, and the number in parentheses (actually their differences) indicate the degrees of the syzygies. This shows that in this case the first syzygies are 8 and of degree 2, while all other syzygies are linear.

This complex confirmed that the syzygies we had obtained via analysis were in fact the only ones one needed. The resolution, however, has some additional interesting features as it indicates a symmetry similar to the one encountered in the study of the Cauchy-Riemann system. Not only there is symmetry in the size of the complex, but the last operator

\frac{∂}{∂q̃} = \frac{∂}{∂x_{0}} + i\frac{∂}{∂x_{1}} + j \frac{∂}{∂x_{2}} − k \frac{∂}{∂x_{3}}

is, apart from some changes in signs, once again a Cauchy-Fueter operator. This showed, among other things, that a duality theorem akin to Köthe’s duality theorem is true in the quaternionic case. That this was the case for one quaternionic variables was already shown in [20], but the theorem in two variables now states that if R and R̃ indicate the sheaf of regular and ∂/∂q̃ regular functions on ℍ^{2} respectively, then there is an isomorphism

(R(K))′ ≅ H_{K}^{3}(ℍ^{2}; R̃).

We then conjectured that a result of this nature could be in fact extended to every dimension, but the next series of computer calculations showed a series of unexpected, and interesting, new features. Further computer calculations showed, for example, that for n = 3 quaternionic variables, the resolution looks like

0 ⟶ R^{8}(−6) ⟶ R^{36}(−5) ⟶ R^{60}(−4) ⟶
⟶ R^{40}(−3) ⟶ R^{12}(−1) ⟶ R^{4} ⟶ M_{3} ⟶ 0

while for n = 4 variables it looks like

0 ⟶ R^{12}(−8) ⟶ R^{80}(−7) ⟶ R^{224}(−6) ⟶ R^{336}(−5) ⟶

1CoCoA is an ongoing special computer system for doing computations in commutative algebra. It is freely available by anonymous ftp from http://cocoa.dima.unige.it

⟶ R^{280}(−4) ⟶ R^{112}(−3) ⟶ R^{16}(−1) ⟶ R^{4} ⟶ M_{4} ⟶ 0

Given the complexity of the computations involved (one sees that the dimension of the matrices grows up to 336), we could not find a machine which could compute the resolution of the Cauchy-Fueter system for higher values of n. But the data we had collected for n = 3 and n = 4 were sufficient to conjecture a few relevant properties that we then set out to demonstrate. Specifically, we conjectured the following facts:

i) The resolutions involve quadratic syzygies at the first step, while only linear syzygies appear at all the other steps.

ii) The length of the resolution of the Cauchy-Fueter system in n quaternionic variables is 2n − 1.

iii) All Ext groups vanish, with the exception of the last one.

iv) It should be possible to find precise formulas for the Betti numbers of the resolutions (the exponents in the resolutions themselves).

At the same time, we quickly realized that the computations we had carried out were also showing some new unexpected facts:

j) The resolutions lack symmetry in the Betti numbers as n grows higher than 2.

jj) The last operator is not necessarily (in fact it is never) a simple variation of the Cauchy-Fueter operator; in general we conjecture that not only variations of the Cauchy–Fueter are involved but, more in general, also operators that are projections of the Cauchy-Fueter operator on suitable subspaces; thus, if a duality theorem can be proved, it will involve a new sheaf, and not just the sheaf R.

jjj) The syzygies which appear at the first step of the resolution cannot always be represented in terms of operations (basically commutation, anti-commutation and conjugation) on quaternions; this conjecture comes from a detailed analysis of those first syzygies, an analysis which we cannot replicate here.

Though most of these conjectures appear natural once one looks carefully at the resolutions described before, the question (crucial for the development of computational algebraic analysis) is how to go about proving them. The central idea is to essentially follow the same lines which guide the construction of CoCoA.

As workers in computational commutative algebra know very well, the most important theoretical development which has allowed the birth of such a discipline was the discovery, in the sixties, of Gröbner Bases, which is due to the doctoral work of Buchberger [8], himself a student of Gröbner (see the volume [9], as well as our own article [42] in it). The idea behind Gröbner Bases (see the excellent and modern treatment in [2]) is to find a way to construct special bases for ideals (and modules) in the ring of polynomials. These special bases are constructed in such a way as to allow direct computations (theoretically highly complex, but in most cases actually accessible) for a large variety of important algebraic objects. It did not take long for this concept to become extremely successful, and many software packages were built around this tool. The two most famous are probably Macaulay2^{2}, and CoCoA. The point we wish to make, however, is that Gröbner Bases are the theoretical instrument which allows these computations. For this reason, we decided that it was through the use of Gröbner Bases that we could hope to demonstrate (or disprove) the conjectures and observations listed above.

2Macaulay2 was developed by D. R. Grayson and M. E. Stillman and is is freely available at http://www.math.uiuc.edu.

In a series of papers (see [1], [7], [3], [4]), we succeeded (together with many collaborators from diverse areas of mathematics) to prove the following general results (which we condensate here in a single theorem for the reader’s ease):

Theorem 4.4. Let M_{n} be the module associated to the Cauchy Fueter system in n > 1 variables. Then its resolution is

0 ⟶ R^{β_{2n−1}}(−2n) ⟶ R^{β_{2n−2}}(−2n + 1) ⟶ …
⟶ R^{β_{3}}(−4) ⟶ R^{β_{2}}(−3) ⟶ R^{4n}(−1) ⟶ R^{4} ⟶ M_{n} ⟶ 0.

In particular

I) The resolution of M_{n} has length 2n − 1;

II) all the maps in the resolutions are linear except the first one.

Moreover:

III) the Betti number β_{r} at the step r is given by

β_{r} = 4\binom{2n − 1}{r}\frac{n(r − 1)}{r + 1};

IV) Ext^{j}(M, R) = 0 for j = 1, …, 2n − 2, Ext^{2n−1}(M, R) ≠ 0;

This result demonstrates that a computational approach to algebraic analysis is possible, and that in fact computational techniques can be used both at the level of experimentation, and at the level of demonstration.

At this point, one may wonder whether more sophisticated situations can be dealt with. In order to justify the choice of our next steps, we need to point out (but see the next section for more details on this) that both the Cauchy-Riemann system, and its quaternionic analogue, the Cauchy-Fueter system, have relevant interest in both classical and quantum physics. In particular the Cauchy-Fueter system is an example of a more general equation of fundamental importance in physics, namely the so called Dirac equation.

The only way to fully appreciate and describe the Dirac equation, however, is to look at it in the framework of Clifford Algebras. We will not dwell on this aspect too long, except to remind the reader that the Clifford Algebra ℝ_{m} on m elements is the real algebra generated by m elements e_{1}, …, e_{m}, together with the Clifford relations e_{i}e_{j} + e_{j}e_{i} = −2δ_{ij}, where δ_{ij} is the Kronecker’ delta. Clearly, first natural examples of such algebras are the algebra ℂ of complex numbers (for m = 1), and the algebra ℍ of quaternions (for m = 2). So the theory of the Cauchy-Riemann and the Cauchy-Fueter systems can be subsumed in a more general theory for the Dirac equation defined through the Dirac operator

∂_{x} := ∑_{i=1}^{m} \frac{∂}{∂x_{i}}

and its variations. The algebraic study of Dirac operators adds an element of further complexity to the picture we have painted, since we can now consider n Clifford variables on the algebra ℝ_{m}, thus requiring our computations to deal with two interacting integers, m and n.

Our first experiments dealt with several cases in which both indices were sufficiently small to allow our machines to perform the computations. As an example, we provide here the resolution (in terms of real coordinates) for the cases n = 3, m = 6.

0 ⟶ R^{8}(−9) ⟶ R^{24}(−8) ⟶ R^{64}(−6) ⟶ R^{48}(−4) ⊕ R^{48}(−5) ⟶
⟶ R^{64}(−3) ⟶ R^{24}(−1) ⟶ R^{8} ⟶ M_{3} ⟶ 0,  (3)

The resolution in this case is particularly significant because of the following result we we proved in [37]

Theorem 4.5. The resolution (3) is the resolution for systems of three Dirac operators in any Clifford Algebra over m ≥ 5 units. All the maps appearing in the complex can be written explicitly in terms of ∂_{x_{1}}, ∂_{x_{2}}, ∂_{x_{3}} since they all come from the so–called radial algebra defining relations

{∂_{x_{i}}, ∂_{x_{j}}}∂_{x_{k}} = ∂_{x_{k}}{∂_{x_{i}}, ∂_{x_{j}}}.

Remark 4.6. When we say that (3) is the resolution in an arbitrary Clifford Algebra ℝ_{m} we mean that the number, and the degrees of the syzygies at each step are always the same. Actually the Betti numbers change according to the dimension m involved but they remain proportional to the ones in (3).

Maybe the most interesting consequence of [37], is that the first syzygies can be written in much simpler terms than we had been able to do in the case of quaternionic variables. This was, to some extent, a rather surprising result, as we had feared that increasing the complexity of the algebraic structure would have resulted in an increasing complexity of the resolutions. A geometric interpretation for this phenomenon was possible, however, and we discovered that what happens in the case of quaternions (where the so-called exceptional syzygies appear), is a consequence of the low dimensionality of the Clifford algebra involved. Thus, we were able to prove the following result:

Theorem 4.7. If the dimension of the Clifford Algebra ℝ_{m} is such that m ≥ 2n, and the first syzygies are quadratic, then the resolution of the Dirac complex in n variables can only have syzygies consisting of radial polynomials. Moreover the complex can be computed in terms of abstract vector derivatives and remains the same for all values m ≥ 2n.

While our study of the situations surrounding the case of Dirac operators is far from concluded, we believe that what we have discussed and presented furnishes sufficient proof of the relevance and success of the method we are employing.

5 Some Physical Application of Algebraic Analysis

In this section we wish to provide the reader with some possible applications of the theory we have so far presented. First of all, we wish to point out that the theory we have developed in the case of the Cauchy–Fueter or Dirac operator operator in the quaternionic and Clifford Algebras case respectively, is already worth of attention. For example, see [6], the regular functions (in the sense of the Cauchy-Fueter operator) are associated to massless right-handed neutrino fields, as this operator is the well known Dirac-Weyl operator acting on spinors over ℝ^{4}. The Cauchy-Fueter complex that we have constructed in Theorem 4.4 has a parallel construction made by Baston in [6] using completely different methods relying on the local twistor theory and on the Penrose transform (version of an integral transform analogue to the Radon transform). The two complexes start with the same operator, have the same length and Betti numbers. It has been shown ([38]) that, at least in some particular cases, the two complexes coincide so it will be interesting to solve the following Research problem 2. Provide a general proof that the two constructions of the quaternionic complex coincide.

The parallelism with the work of Baston leads us to another important consideration: the study of Cauchy-Fueter and Dirac operators fits into the more general topic of the study of invariant first order operators, see [39]. The Dirac operator is (up to a multiple) the unique first order, elliptic, conformally invariant operator with values in a basic irreducible spinor representation. When considering systems of several operators the situation becomes more complicated: for example the group of transformations leaving invariant the solutions of the system of n > 1 Cauchy-Fueter operators is bigger than the product of the conformal maps in the single variables. This makes the study particularly interesting because it has been shown that one has to consider a suitable subset of the group of automorphisms of the quaternionic projective space.

It is not clear, so far, what group needs to be considered in the case of several Dirac operators so it will be interesting to study the complex of several Dirac operators from this point of view and then to carry out the comparison with what we have proved so far. An analysis from these two different perspectives should clarify the presence of the exceptional syzygies in the case of Clifford Algebras with low dimension (compare Theorem 4.7). This suggests the following Research problem 3. Study the invariance of the system of several Dirac operators in a Clifford Algebra. Research problem 4. Compare the resulting complex with the one obtained in [37].

As further applications of the algebraic techniques we have discussed, we wish to show how the maps in the resolution of an operator associated to a physical system can be interpreted in terms of conservation laws (see [13]). The underlying idea is that when we consider a physical system, the maps in the resolution should have a physical interpretation.

To start with, we will consider the Maxwell system. Let E = [E_{1}, E_{2}, E_{3}], B = [B_{1}, B_{2}, B_{3}] be the electric and the magnetic fields and ρ^{e}, ρ^{m} the density of electric and magnetic charges, while j^{e}, j^{m} are the electric and magnetic currents, so we have

\{ \begin{matrix}∇ · E = ρ^{e} \\ ∇ · B = ρ^{m} \\ −∇ × E − ∂_{t}B = j^{m} \\ ∇ × B − ∂_{t}E = j^{e}. \end{matrix}  (4)

If we set ρ^{m} = 0 and j^{m} = 0 we get the usual Maxwell system and standard calculations show that it implies the conservation of charge given by:

∂_{t}ρ^{e} + ∇ · j^{e} = 0.  (5)

In the general case, we will consider system (4) from an algebraic point of view. We set

F = [E_{1}, E_{2}, E_{3}, B_{1}, B_{2}, B_{3}]^{t},  (6)
G = [ρ^{e}, ρ^{m}, j_{1}^{m}, j_{2}^{m}, j_{3}^{m}, j_{1}^{e}, j_{2}^{e}, j_{3}^{e}]^{t}

so we can rewrite the Maxwell equations in matrix form as

P_{M}(D)F = G

where the matrix of differential operators P_{M}(D) is defined by

P_{M}(D) = [ \begin{matrix}∂_{x} & ∂_{y} & ∂_{z} & 0 & 0 & 0 \\ 0 & 0 & 0 & ∂_{x} & ∂_{y} & ∂_{z} \\ 0 & ∂_{z} & −∂_{y} & −∂_{t} & 0 & 0 \\ −∂_{z} & 0 & ∂_{x} & 0 & −∂_{t} & 0 \\ ∂_{y} & −∂_{x} & 0 & 0 & 0 & −∂_{t} \\ −∂_{t} & 0 & 0 & 0 & −∂_{z} & ∂_{y} \\ 0 & −∂_{t} & 0 & ∂_{z} & 0 & −∂_{x} \\ 0 & 0 & −∂_{t} & −∂_{y} & ∂_{x} & 0 \end{matrix}] .

The resolution of the module M = R^{6}/P_{M}^{t}R^{8}, where R = ℂ[t, x, y, z], turns out to be:

0 ⟶ R^{2}(−2) ⟶^{P_{1}^{t}} R^{8}(−1) ⟶^{P_{M}^{t}} R^{6} ⟶ M ⟶ 0.

Looking at the Betti numbers and at the degree of the maps, we deduce the existence of two syzygies of degree 1. The matrix operator associated to the first syzygies can be written in the form

P_{1}(D) = [ \begin{matrix}0 & ∂_{t} & ∂_{x} & ∂_{y} & ∂_{z} & 0 & 0 & 0 \\ ∂_{t} & 0 & 0 & 0 & 0 & ∂_{x} & ∂_{y} & ∂_{z} \end{matrix}] ,  (7)

and can be read in terms of compatibility conditions as follows:

Theorem 5.1. The non homogeneous Maxwell system

P_{M}(D)F = G

has a solution F ∈ S^{6} if and only if G ∈ S^{8} satisfies

P_{1}(D)G = 0.  (8)

In physical terms this statement translates into the next

Proposition 5.2. In the physical case in which there are no magnetic monopoles, (8) gives only the condition

∂_{t}ρ^{e} + ∇ · j^{e} = 0,

while in the case there are magnetic monopoles and magnetic currents the solvability conditions (8) become

\{ \begin{matrix}∂_{t}ρ^{e} + ∇ · j^{e} = 0 \\ ∂_{t}ρ^{m} + ∇ · j^{m} = 0. \end{matrix}

Note that it is possible to describe the Ext-modules:

Proposition 5.3. The characteristic variety of Ext^{1}(M, R), is given by

(t^{2} − x^{2} − y^{2} − z^{2})^{2} = 0

so that Ext^{1}(M, R) ≠ 0, moreover Ext^{2}(M, R) = 0.

The proof is based on a quick computation with CoCoA and shows that the system is not overdetermined. In other words, compact singularities cannot be eliminated.

From a physical point of view we can say that the possible existence of compact singularities may be interpreted as the presence of localized charges which generate the electromagnetic field. The explicit description of the characteristic variety also allows to prove the following fact:

Proposition 5.4. Let Ω be an open set in R^{4} and let x be a point in Ω. Then every solution of the Maxwell system in Ω\x, whose components can be extended as distributions to all of Ω, is a distribution solution to the Maxwell system to all of Ω.

More in general, we can consider the Proca equations (originally introduced in [34]) that generalize Maxwell field to the case of massive spin 1 particles. An example is represented by the particles W^{±} and Z related to the electroweak interaction. The Proca equations can be written in the following form, where we assume the absence of magnetic monopoles:

\{ \begin{matrix}∇ · E = ρ^{e} + m^{2}A_{0} \\ ∇ · B = 0 \\ ∇ × E = −∂_{t}B \\ ∇ × B = ∂_{t}E + m^{2}A + j^{e} \\ B = ∇ × A \\ E = −∇A^{0} − ∂_{t}A. \end{matrix}  (9)

Remark 5.5. In this case, we deduce the following conservation law:

∂_{t}(m^{2}A^{0} + ρ^{e}) + ∇ · (m^{2}A + j^{e}) = 0  (10)

The main difference with respect to the electromagnetic field is that the four current (ρ^{e}, j^{e}) is not necessarily conserved. This fact depends on the particular nature of the source (see [23]). If we assume that the source (A^{0}, A) is conservative, that is

∂_{t}A^{0} + ∇ · A = 0  (11)

we get from (10) the conservation law of the electric charge.

Let us consider now the Proca system written in a more general form in which there are no homogeneous equations. From the mathematical point of view, we can look at

\{ \begin{matrix}∇ · E − m^{2}A_{0} = ρ^{e} \\ ∇ · B = ρ^{m} \\ −∇ × E − ∂_{t}B = j^{m} \\ ∇ × B − ∂_{t}E − m^{2}A = j^{e} \\ B − ∇ × A = B^{ext} \\ E + ∇A^{0} + ∂_{t}A = E^{ext} \end{matrix}  (12)

where we introduce, besides electric and magnetic sources, also E^{ext}, B^{ext} that we call external fields. We want to study the resolution and the compatibility conditions that arise in this case.

We rewrite the system (12) in matrix form as follows

P_{P}(D)F = G

where we have set

F = [E_{1}, E_{2}, E_{3}, B_{1}, B_{2}, B_{3}, A_{0}, A_{1}, A_{2}, A_{3}]^{t},  (13)
G = [ρ^{e}, ρ^{m}, j_{1}^{m}, j_{2}^{m}, j_{3}^{m}, j_{1}^{e}, j_{2}^{e}, j_{3}^{e}, B_{1}^{ext}, B_{2}^{ext}, B_{3}^{ext}, E_{1}^{ext}, E_{2}^{ext}, E_{3}^{ext}]^{t},

and the matrix of differential operators P_{P}(D) is defined by

[\begin{matrix}∂_{x} & ∂_{y} & ∂_{z} & 0 & 0 & 0 & −m^{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & ∂_{x} & ∂_{y} & ∂_{z} & 0 & 0 & 0 & 0 \\ 0 & ∂_{z} & −∂_{y} & −∂_{t} & 0 & 0 & 0 & 0 & 0 & 0 \\ −∂_{z} & 0 & ∂_{x} & 0 & −∂_{t} & 0 & 0 & 0 & 0 & 0 \\ ∂_{y} & −∂_{x} & 0 & 0 & 0 & −∂_{t} & 0 & 0 & 0 & 0 \\ −∂_{t} & 0 & 0 & 0 & −∂_{z} & ∂_{y} & 0 & −m^{2} & 0 & 0 \\ 0 & −∂_{t} & 0 & ∂_{z} & 0 & −∂_{x} & 0 & 0 & −m^{2} & 0 \\ 0 & 0 & −∂_{t} & −∂_{y} & ∂_{x} & 0 & 0 & 0 & 0 & −m^{2} \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & ∂_{z} & −∂_{y} \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & −∂_{z} & 0 & ∂_{x} \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & ∂_{y} & −∂_{x} & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & ∂_{x} & ∂_{t} & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & ∂_{y} & 0 & ∂_{t} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & ∂_{z} & 0 & 0 & ∂_{t} \end{matrix}] .

The computation of the resolution with CoCoA gives the following result:

0 ⟶ R(−3) ⟶^{P_{2}^{t}} R^{5}(−2) ⟶^{P_{1}^{t}} R^{14}(−1) ⟶^{P_{P}^{t}} R^{10} ⟶ M ⟶ 0

where R = ℂ[t, x, y, z, m]. The surprising fact is that in addition to the first syzygies, i.e. some compatibility conditions on the data of system (12), we also have second syzygies, i.e. compatibility conditions on the data of the inhomogeneous system associated to the matrix P_{1}. We do not write here the matrices appearing in the resolutions: what is important is that, looking at these relations, one discovers that they can be grouped into three classes: two scalars and one vectorial equations. The result is summarized in the following theorem.

Theorem 5.6. The non homogeneous Proca system

P_{P}(D)F = G

has a solution F ∈ S^{10} if and only if G ∈ S^{14} satisfies P_{1}(D)G = 0, i.e.

\{ \begin{matrix}ρ^{m} − ∇ · E^{ext} = 0 \\ −j^{m} − ∂_{t}B^{ext} − ∇ × E^{ext} = 0 \\ ∂_{t}ρ^{m} + ∇ · j^{m} = 0. \end{matrix}  (14)

This last system represents three conservations laws, but note the fundamental fact that those laws depend only on the presence of magnetic charge and currents. In fact, if one supposes E^{ext} = B^{ext} = 0 then the first two equations become j^{m} = ρ^{m} = 0 and the third equation is trivially an identity. The existence of external fields makes the system non trivial. If we now consider the non-homogeneous version of the system (14)

P_{1}(D)G = H  (15)

by considering the datum H = [ℓ_{1}, L, ℓ_{2}], we get

Theorem 5.7. System (15) has a solution G ∈ S^{14} if and only if H ∈ S^{5} satisfies

P_{2}(D)H = 0,

i.e. the datum must satisfy the equation

−∂_{t}ℓ_{1} + ∇ · L + ℓ_{2} = 0.  (16)

We can interpret −L as a new magnetic current density, ℓ_{1} the relative magnetic monopoles density and ℓ_{2} the source of the field. If ℓ_{2} = 0 equation (16) represents a conservation law for the magnetic density ℓ_{1}, while if ℓ_{2} ≠ 0 the differential equation (16) gives the evolution of [ℓ_{1}, L] in terms of the source ℓ_{2} in order for the system (14) to have a solution.

Note that in the case of the Maxwell system we only have first syzygies so that the nonhomogeneous system

\{ \begin{matrix}∂_{t}ρ^{e} + ∇ · j^{e} = ℓ \\ ∂_{t}ρ^{m} + ∇ · j^{m} = ℓ′ \end{matrix}

can be solved without conditions on ℓ and ℓ′. Even in the case of Proca equations, the system is not overdetermined. In fact the matrix P_{P} is of maximal rank but the greatest common divisor of its minors of maximal rank is not equal to 1. More precisely, we have

Proposition 5.8. The characteristic variety of Ext^{1}(M, R), in case of system (12) is given by

m^{2}(t^{2} − x^{2} − y^{2} − z^{2} − m^{2})^{3} = 0

so that Ext^{1}(M, R) ≠ 0, moreover Ext^{2}(M, R) ≠ 0, while Ext^{3}(M, R) = 0.

Note that the characteristic variety of Ext^{1}(M, R) is algebraic and is defined by the ideal generated by the greatest common divisor of the minors of maximum order. This (non reduced) variety is the light cone t^{2} − x^{2} − y^{2} − z^{2} = 0 for the Maxwell system, and the hyperboloid t^{2} − x^{2} − y^{2} − z^{2} − m^{2} = 0 for the Proca system.

Remark 5.9. According to Remark 5.5, we can consider the Proca system under the assumption that the field (A_{0}, A) is conserved, i.e. supposing that

∂_{t}A_{0} + ∇ · A = 0.

The non homogeneous system (12) can be rewritten taking into account this new condition, adding the equation

∂_{t}A_{0} + ∇ · A = A^{ext},

where A^{ext} is a given datum. The system is represented by a 15 × 10 matrix whose resolution is

0 ⟶ R(−3) ⟶^{P_{2}^{t}} R^{6}(−2) ⟶^{P_{1}^{t}} R^{15}(−1) ⟶^{P_{P}^{t}} R^{10} ⟶ M ⟶ 0

The compatibility conditions of this new system are given by (14) and the relation

A^{ext} + ∂_{t}ρ^{e} + ∇ · j^{e} = 0.

According to the physical considerations in Remark 5.5, when A^{ext} = 0 we get, in particular, the conservation law of the electric charge.

Remark 5.10. To conclude this section, we wish to describe the characteristic variety of the Maxwell and Proca systems that are essential to write the integral representation of their solutions by the Ehrenpreis–Palamodov’s Fundamental Principle. By Remark 2.11, the characteristic variety V_{M} of the Maxwell system can be obtained as the vanishing set of the ideal generated by the 6 × 6 minors of the matrix P_{M}, i.e. the ideal generated by the 10 polynomials of the type

(x^{2} + y^{2} + z^{2} − t^{2})^{2}χ_{i}(x, y, z, t), i = 1, …, 10

where χ_{i} is a degree two monomial in the variables x, y, z, t. The characteristic variety V_{P} of the Proca system can be written in a similar way, as the vanishing set of the ideal generated by the 10 × 10 minors of the matrix P_{P} associated to the Proca system (12), i.e. the ideal generated by the 35 polynomials of the type

(x^{2} + y^{2} + z^{2} − t^{2} + m^{2})^{3}η_{i}(x, y, z, t), i = 1, …, 35

where η_{i} is a monomial of degree less than or equal three in the variables x, y, z, t.

The discussion is this section shows to what extent the computational algebraic analysis is powerful even when treating the case of systems with a physical meaning: from one side, it is possible to rediscover very well known facts (like some conservation laws) but also to envision new features of the system.

6 Some Directions for Further Research

The theory which we have developed in these last few years, and which we have summarily described in this paper, is still in flux and as we solve problems, new challenges naturally surface. We have already included in the body of this paper some of the research problems we need to address. We wish to conclude the paper by describing a few more directions for further work. The first is related to the parallelism between the Koszul and the Dolbeault complexes which we discussed in section 3. When one considers Dirac complexes, there are no differential forms defined in a natural way. In particular, since we know that the first syzygies are quadratic, we cannot expect to built a theory of degree one differential forms. This has lead us, in [36], to construct a new differential complex based on what we called “megaforms” in which a basis exists consisting of degree one forms together with a special basis for degree two forms. The theory of this complex is quite interesting, but further work will be needed beyond what was done in [36].

A second direction has to do with the computational aspects of the theory. We have shown that to treat systems of differential operators in a Clifford algebra, it is necessary to rewrite the systems in terms of the real components over the various units. As the dimension of the algebra grows, the number of these components grows exponentially. It will be useful to have an algorithm taking into account the relations e_{i}e_{j} + e_{j}e_{i} = −2δ_{ij} on the units {e_{i}} of the Clifford algebra, so that one could consider Clifford equations instead of their real counterparts.

Moreover, it will be necessary to find good algorithms to explicitly compute the functors Ext and Hom given a specific system. Those functors are applied to finitely generated modules over ring of polynomials so the theory of Gröbner basis is the suitable tool to describe them as modules. The idea is to construct the presentation of the modules Ext and Hom as quotients of a suitable free module. This has been done in the case of the module Ext and homogeneous systems in [15]. Using a suitable trick to handle the shifts, the algorithm can be used also in the non homogeneous case. Our hope is to optimize the algorithms for Ext and to create one to compute Hom.

At the end of section 4 we have mentioned the fact that all the maps in the complex of several Dirac operators in a Clifford algebra ℝ_{m} can be explicitly written in terms of the radial algebra relations, provided that the dimension of ℝ_{m} is at least twice the number of operators. If the dimension is lower, as in the quaternionic case, than this is no more true and exceptional syzygies can appear. The presence of those syzygies entails also counterexamples to the Fischer decomposition as done in [37]. We can ask if it is possible to write a suitable Fischer decomposition for regular functions of quaternionic variables. More precisely, the problem is to find suitable operator(s) and polynomials such that a regular function can be decomposed as the sum of a function in the kernel of the chosen operators and a function in the ideal of the chosen polynomials. We conjecture that such a decomposition can exists at least in the case of regular functions of one quaternionic variable.

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