| This dissertation is the result of my research on the algebraic analysis of linear constant coefficients differential operators, and its computational aspects. It is the natural continuation of the work I initiated with my Laurea Thesis [27] where I studied the computation of the Ext modules using Grobner Bases and their analytical meaning for the function theory of kernels of linear constant coefficient partial differential operators. As the research on Dirac operators developed, new theoretical aspects have emerged and new relations with computer algebra and Grobner Basis theory have been considered. In this dissertation I will provide some background on the algebraic analysis of such operators, and will then summarize the latest results on this topic before proceeding to a detailed exposition of my own results. I will devote particular attention to the use of Grobner Bases and the experiments conducted with some computer algebra software packages such as CoCoA [21], Macaulay2 [44] and Singular [46]. Part of the work is also the CoCoA code I wrote to perform experiments and to implement the algorithms for the explicit calculation of the mathematical objects involved in the Fourier analysis of the operators. |
