Rational expression simplification with polynomial side relations

The goal of this thesis is to develop generic algorithms for computing in polynomial quotient rings and their fields of fractions. We present two algorithms for simplifying rational expressions over k[xl, . . . , x,]/ I. The first algorithm uses Groebner bases for modules to compute an equivalent expression whose largest term is minimal with respect to a given monomial order. The second algorithm solves systems of linear equations to find equivalent expressions and conducts a brute force search to find an expression of minimal total degree.