PART II: Research Proposal Algorithms for the Simplification of Algebraic Formulae

Computer algebra (CA) or symbolic computation, as my field is known by, consists of two principal research activities, namely, the design, implementation and analysis of algebraic algorithms and, secondly, the design of programming languages, data structures and sofware environments for implementing these algorithms. The research problem that I propose to work on is the simplification problem. Let f be an algebraic expression (or formula) in n unknowns (x1, ..., xn). The simplification problem is to compute amongst all expressions which are equal to f (at all values of the unknowns) one which is simplest, that is, of smallest size according to some metric. An important special case of the simplification problem is the zero-equivalence problem, that is, testing whether f(x1, ..., xn) = 0. In terms of importance, I claim that simplification is the single most important functionality provided by a computer algebra system for general users, i.e., scientists and engineers using such systems to compute and manipulate formulae. To simplify an algebraic expression, some of the tools that we use include solutions to the following computational problems: (i) computing the greatest common divisor (GCD) of two polynomials, (ii) factoring a polynomial into irreducible factors, (iii) decomposing a polynomial, (iv) computing a Gröbner basis for a polynomial ideal, and (v) computing a prime(ary) decomposition of a polynomial ideal. We also use tools from linear algebra. The long term goal of this proposal is to design effective algorithms and build the best software system for solving the simplification problem. I also propose to do basic research on (i), in particular, computing GCDs of multivariate polynomials over algebraic function fields and triangular sets using modular methods.