Termination of generator enumeration for A(I) when finitely generated

Determine whether there exists a general algorithmic procedure that, given an arbitrary ideal I ⊆ K[x_1,…,x_n,y_1,…,y_m], enumerates generators of the algebra of separated elements A(I) and is guaranteed to terminate whenever A(I) is finitely generated.

Background

The paper studies the algebra A(I) of separated elements for ideals I in multivariate polynomial rings K[X,Y], where separated elements are of the form f−g with f∈K[X] and g∈K[Y]. For principal ideals not contained in K[X]∪K[Y], A(I) is shown to be simple and its generator can be obtained via a reduction to a bivariate problem. For ideals of dimension zero, a finite generating set can be computed.

For arbitrary ideals, A(I) may not be finitely generated. The authors propose procedures to enumerate generators of A(I) but cannot ensure termination in the general case. The open question concerns the existence of an enumeration procedure that halts precisely when A(I) is finitely generated, which would bridge the gap between non-terminating enumeration and guaranteed finite computation in favorable cases.