Greatest common divisors of polynomials given by straight-line programs

Erich Kaltofen
1988 Journal of the ACM  
Algorithms on multivariate polynomials represented by straight-line programs are developed. First it is shown that most algebraic algorithms can be probabilistically applied to data that is givenb ya straight-line computation. Testing such rational numeric data for zero, for instance, is facilitated by random evaluations modulo random prime numbers. Then auxiliary algorithms are constructed that determine the coefficients of a multivariate polynomial in a single variable. The first main result
more » ... first main result is an algorithm that produces the greatest common divisor of the input polynomials, all in straight-line representation. The second result shows howtofind a straight-line program for the reduced numerator and denominator from one for the corresponding rational function. Both the algorithm for that construction and the greatest common divisor algorithm are in random polynomial-time for the usual coefficient fields and output a straight-line program, which with controllably high probability correctly determines the requested answer.T he running times are polynomial functions in the binary input size, the input degrees as unary numbers, and the logarithm of the inverse of the failure probability.T he algorithm for straight-line programs for the numerators and denominators of rational functions implies that every degree bounded rational function can be computed fast in parallel, that is in polynomial size and poly-logarithmic depth.
doi:10.1145/42267.45069 fatcat:hjux5gwgybe5nikgp3nozu7zzq