Computing with polynomials given by black boxes for their evaluations: greatest common divisors, factorization, separation of numerators and denominators

Algorithms are developed that adopt a novel implicit representation for multivariate polynomials and rational functions with rational coefficients, that of black boxes for their evaluation. It is shown that within this evaluation-box representation, the polynomial greatest common divisor and factorization problems as well as the problem of extracting the numerator and denominator of a rational function can be solved in random polynomial time in the usual parameters. Since the resulting evaluation programs for the goal polynomials can be converted efficiently to sparse format, solutions to sparse problems such as the sparse ration interpolation problem follow as a consequence