Computing the minimal number of equations defining an affine curve ideal-theoretically

There is an algorithm which computes the minimal number of generators of the ideal of a reduced curve C in affine n-space over an algebraically closed field K, provided C is not a local complete intersection. The existence of such an algorithm follows from the fact that given , there exists , such that if is a height n−1 radical ideal in K[X1,…,Xn], generated by polynomials of degree at most d, then admits a set of generators of minimal cardinality, with each generator having degree at most d′, except possibly when is an (unmixed) local complete intersection.