Algorithms for Determining Birationality of Parametrization of Affine Curves

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Let k be an arbitrary field, and C be a curve in Aⁿ 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 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=(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.