Sur une conjecture de Schinzel

TL;DR

This paper provides a new, explicit proof of Schinzel's conjecture concerning the intersection of certain subvarieties with a one-dimensional torus, utilizing geometric and height bounds techniques.

Contribution

It introduces an explicit proof of Schinzel's conjecture using geometric Bézout's theorem and height bounds, extending previous analogous results on elliptic curves.

Findings

Explicit bounds depending on height and degree of the variety

First explicit proof of Schinzel's conjecture in this context

Extension of methods from elliptic curves to higher-dimensional tori

Abstract

We give a new proof of a conjecture of Schinzel on the intersection of a subvariety of codimension at least 2 in a power of the multiplicative group with a torus of dimension 1. The proof rests on a geometric B\'ezout's theorem of P. Philippon and on lower bounds for the height of the first author, S. David and E. Viada. It gives for the first time an explicit result, depending on the height and degree of the variety. It is inspired on a similar statement on products of elliptic curves, by S. Checcoli, F. Veneziano and E. Viada.