For a given ideal <?TeX $I\subseteq \mathbb {K}[x_1,\dots,x_n,y_1,\dots,y_m]$?> Math 1 in a polynomial ring with n + m variables, we want to find all elements that can be written as f − g for some <?TeX $f\in \mathbb {K}[x_1,\dots,x_n]$?> Math 2 and some <?TeX $g\in \mathbb {K}[y_1,\dots,y_m]$?> Math 3 , i.e., all elements of I that contain no term involving at the same time one of the x1, …, xn and one of the y1, …, ym. For principal ideals and for ideals of dimension zero, we give a algorithms that compute all these polynomials in a finite number of steps.

We present an algorithm which for any given ideal I ⊆ K[x, y] finds all elements of I that have the form f(x) - g(y), i.e., all elements in which no monomial is a multiple of xy.

We consider the problem of learning the topology of a graph from a given set of smooth graph signals. We construct a weighted adjacency matrix that best explains the data in the sense of achieving the smallest graph total variation. For the case of noisy measurements of the graph signals we propose a scheme that simultaneously denoises the signals and learns the graph adjacency matrix. Our method allows for a direct control of the number of edges and of the weighted node degree. Numerical experiments demonstrate that our graph learning scheme is well suited for community detection.