On the Problem of Separating Variables in Multivariate Polynomial Ideals

For a given ideal I in K[x_1,…,x_n,y_1,…,y_m] in a polynomial ring with n+m variables, we want to find all elements that can be written as f-g for some f in K[x_1,…,x_n] and some g in K[y_1,…,y_m], i.e., all elements of I that contain no term involving at the same time one of the x_1,…,x_n and one of the y_1,…,y_m. For principal ideals and for ideals of dimension zero, we give a algorithms that compute all these polynomials in a finite number of steps.