A Rational Function Decomposition Algorithm by Near-separated Polynomials

In this paper we present an algorithm for decomposing rational functions over an arbitrary coefficient field. The algorithm requires exponential time, but is more efficient in practice than the previous ones, including the polynomial time algorithm. Moreover, our algorithm is easier to implement. We also present some applications of rational function decomposition: (1) faithful re-parameterizing unfaithfully parameterized curves, (2) computing intermediate fields in a simple purely transcendental field extension , and (3) providing a birationality test for subfields of (x). Several examples are computed using an implementation of our algorithm using MAPLE V.