An Algorithm to Compute the Adjoint Ideal of an Affine Plane Algebraic Curve

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Let \({\fancyscript{C}}\) be an irreducible affine plane algebraic curve defined by a polynomial \({f \in K[x, y]}\). Assume that f is monic with respect to y and that the function field \({K(\fancyscript{C})}\) of the curve is separable over K(x). In this paper we give a simple algorithm to compute a lexicographic Groebner basis of the adjoint ideal \({{\rm Adj}(\fancyscript{C})}\) of \({\fancyscript{C}}\) from a basis, as a K[x]-module, of the integral closure \({K[\overline{\fancyscript{C}}]}\) of the coordinate ring \({K[\fancyscript{C}]}\).