In the space of polynomials in two variables \(\mathbb{C}[x,y]\) with complex coefficients we let the group of automorphisms of the affine plane \(\mathbb{A}^2_{\mathbb{C}}\) act by composition on the right. In this paper we investigate the geometry of the orbit space.

We associate a graph with each polynomial in two variables that encodes part of its geometric properties at infinity; we define a partition of \(\mathbb{C}[x,y]\) imposing that the polynomials in the same stratum are the polynomials with a fixed associated graph. The graphs associated with polynomials belong to certain class of graphs (called behaviour graphs), that has a purely combinatorial definition. We show that any behaviour graph is actually a graph associated with a polynomial. Using this we manage to give a quite precise geometric description of the subsets of the partition.

We associate a moduli functor with each behaviour graph of the class, which assigns to each scheme \(T\) the set of families of polynomials with the given graph parametrized over \(T\). Later, using the language of groupoids, we prove that there exists a geometric quotient of the subsets of the partition associated with the given graph by the equivalence relation induced by the action of Aut\((\mathbb{C}^2)\). This geometric quotient is a coarse moduli space for the moduli functor associated with the graph. We also give a geometric description of it based on the combinatorics of the associated graph.

The results presented in this memoir need the development of a certain combinatorial formalism. Using it we are also able to reprove certain known theorems in the subject.