We introduce the harmonic decomposition for polynomials (and analytic functions) and then we observe how some linear operators (multiplication, differentiation, inverse Laplacian) act on it. Unexpectedly, an efficient way to represent the action of these operators is a graph with vertices indexed by $\mathbb Z^2$. This representation transforms convergence issues into the study of paths in such graphs. We will apply these tools to the study of two PDEs involving the Laplacian: the construction of a canonical fundamental solution for a second-order differential operator, and a canonical bijection between minimal graphs and harmonic functions.

This is an ongoing project with F. Franceschini.