Rational Simplification Modulo a Polynomial Ideal

We present two algorithms for simplifying rational expressions modulo an ideal of the polynomial ring k[x1,..., xn]. The first method generates the set of equivalent expressions as a module over k[x1,..., xn] and computes a reduced Gröbner basis. From this we obtain a canonical form for the expression up to our choice of monomial order for the ideal. The second method constructs equivalent expressions by solving systems of linear equations over k, and conducts a global search for an expression with minimal total degree. Depending on the ideal, the algorithms may or may not cancel all common divisors. We also provide some timings comparing the efficiency of the algorithms in Maple