Algorithms and applications of comprehensive Groebner bases.

Abstract

In this dissertation we study several improvements to algorithms used to generate comprehensive Groebner bases of Volker Weispfenning (7). Comprehensive Groebner bases are bases for ideals in the ring of polynomials in several variables whose coefficients are polynomials in several symbolic parameters over a given field. These bases have the fundamental property that, for any possible assignment (specialization) of the field elements for the parameters, the comprehensive Groebner basis generators become generators for a usual Groebner basis for the ideal of polynomials with coefficients in the field. Chapter 1 gives the necessary background for understanding the basic construction of comprehensive Groebner bases. We show how it is also possible to construct these bases for ideals in the ring of polynomials whose coefficients are rational functions of the symbolic parameters. We amplify the description of assigning "colors" to coefficients as given in (7). These assignments are used to establish criteria to determine the effect specialization will bear upon a given coefficient of a given polynomial. We also amplify the constructions for S-polynomial and normal form computations in this realm. In Chapter 2, we present several modifications to the algorithms in Chapter 1. These modifications allow for more efficient machine computations and yield simpler output. We show a new design methodology for assignments of "colors" that allows for more readable and useful output. We encode this methodology in our "saturated form" of a condition which makes good use of Groebner basis theory and saturations of ideals. This new design allows for sharper precision when working with comprehensive Groebner bases. In particular, we demonstrate how this design eliminates all unnecessary "virtual" Buchberger algorithms as described in (1). In Chapter 3, we show the precision of our new design in examples from several areas of commutative algebra. We demonstrate the simplicity that may be attained in examples from Automatic Geometric Theorem Proving, and in the study of parametric varieties. Also, we embed our new design in an algorithm that has potential use in the study of dimension bounds of parametric varieties.