Algorithms for Determining Rationality of Parametric Affine Curves

Let f1, . . . , fn ∈ k[t], where k is an arbitrary field. A necessary and sufficient condition for the two function fields k(t) and k(f1, . . . , fn) to be same is developed in terms of zero-dimensionality of an ideal in the bivariate polynomial ring k[s, t]. Since zero-dimensionality of such an ideal can be readily determined by a Gröbner basis computation, this gives an algorithm that determines if the curve C ⊂ Ank given parametrically by x1 = f1, . . . , xn = fn is rational. We also develop an algorithm that determines if k[t] and k[f1, . . . , fn] are same, by which we get an algorithm that determines if C is isomorphic to the affine line. 1991 Mathematics Subject Classification: 13P10