An introduction to computational algebraic analysis (Q1878649)

The authors present a new field of algebraic analysis, which they call computational algebraic analysis. A part of the theory of several complex variables can be interpreted in terms of algebraic analysis. Some systems of differential equations of physical interest are studied with the methods of this new field. Euler's fundamental principle is considered the first result of algebraic analysis. It states that, if \(P\) is a complex polynomial in one variable \(z\) and \(P( d/dx) \) is the differential operator obtained from \(P\) by formally replacing \(z\) by the derivative \(d/dx,\) then, in order to describe the kernel of the operator \(P( d/dx) ,\) we need to find the zeroes of the polynomial \(P\). This can be extended to polynomials of several complex variables. Let \(R\) be the ring of complex polynomials in \(n\) variables \((z_{1},z_{2},\dots,z_{n}) =z.\) If \(z\) is replaced by \(D=( \partial/\partial x_{1},\dots,\partial /\partial x_{n}) \), then \(R\) may be regarded as the ring of symbols of linear partial differential operators with constant coefficients. The study of solutions of a general system of linear partial differential equations with constant coefficients can be reduced to the algebraic task of classifying the morphisms from a certain module \(M\) to the space in which one wants to find the solutions. According to the so-called Hilbert's syzygy theorem, one can always find a finite resolution of the module \(M.\) The maps which arise in these resolutions are known as the syzygies of \(M\). Another important result which is recalled here is the fundamental principle of Ehrenpreis-Palamodov, which allows to write in an explicit way the general solution of a homogeneous system of differential equations with constant coefficients. The authors show how the theory of several complex variables can be reconstructed as an application of the algebraic analysis. For example, the study of holomorphic functions is equivalent to the study of the solutions of the Cauchy-Riemann system and for this, it is necessary to construct an associated Koszul complex. The functions defined on the space of quaternions was first analyzed by Fueter. The quaternionic analogue of the Cauchy-Riemann system is the Cauchy-Fueter system. In this paper, a Cauchy-Fueter system is associated, \(8\) syzygies are constructed and a simple quaternionic interpretation is given. A more general equation than Cauchy-Fueter system is the Dirac equation, which is very important in Physics. This, together with the Clifford Algebras are presented and used here too. Some physical applications of algebraic analysis are given. Among them, the Proca equations, which generalize the Maxwell field. Some open problems are proposed for further research.