A short note on Manin–Mumford

Abstract

We give a short proof of Manin–Mumford in the multiplicative group based on the pigeon-hole principle and a simple structure theorem for tori contained in subvarieties of [Formula: see text] due to Bombieri and Gubler. The arguments appear to be new and are applicable in other situations. Our proof is quite elementary and we compare it to previous proofs that rely on machinery such as intersection theory or [Formula: see text]-minimality.