Guessing a Conjecture in Enumerative Combinatorics and Proving It with a Computer Algebra System

We present a theorem-proving experiment performed with a computer algebra system. It proves a conjecture about the general pattern of the generating functions counting rooted maps of given genus. These functions are characterized by a complex non-linear differential system between generating functions of multi-rooted maps. Establishing a pattern for these functions requires a sophisticated inductive proof. Up to now these proofs were made by hand. This work is the first computer proof of this kind of theorem. Symbolic computations are performed at the same abstraction level as the handmade proofs, but with a computer algebra system. Generalizing this first success may significantly help solving algebraic problems in enumerative combinatorics.