Algorithms for Determining Birationality of Parametrization of Affine Curves

Let k be an arbitrary field, and C be a curve in A defined parametrically by x1 = f1(t), . . . , xn = fn(t), where f1, . . . , fn ∈ k[t]. 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 a derived ideal in the bivariate polynomial ring k[s, t]. Since zerodimensionality of such an ideal can be readily determined by a Gröbner basis computation, this gives an algorithm that determines if the parametrization ψ = (f1, . . . , fn) : A → C is a birational equivalence. 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 the parametrization ψ = (f1, . . . , fn) : A → C is an isomorphism. We include some computational examples showing the application of these algorithms. 1991 Mathematics Subject Classification: 13P10