Algorithms for Determining Birationality of Parametrization of Affine Curves

Abstract

Let $k$ be an arbitrary field, and C be a curve in A^n defined parametrically by x_1=f_1(t),...,x_n=f_n(t), where f_1,...,f_n\in k[t]. A necessary and sufficient condition for the two function fields k(t) and k(f_1,...,f_n) to be same is developed in terms of zero-dimensionality of a derived ideal in the bivariate polynomial ring k[s,t]. Since zero-dimensionality of such an ideal can be readily determined by a Groebner basis computation, this gives an algorithm that determines if the parametrization ψ=(f_1,...,f_n): A --> C is a birational equivalence. We also develop an algorithm that determines if k[t] and k[f_1,...,f_n] are same, by which we get an algorithm that determines if the parametrization ψ=(f_1,...,f_n): A --> C is an isomorphism. We include some computational examples showing the application of these algorithms.