An arithmetic Bern\v{s}tein-Ku\v{s}nirenko inequality

TL;DR

This paper establishes an upper bound for the height of isolated zeros of Laurent polynomial systems over adelic fields, extending the classical Bernstein-Kushnirenko theorem into an arithmetic context using intersection theory.

Contribution

It introduces an arithmetic analogue of the Bernstein-Kushnirenko theorem, providing a new height bound expressed via mixed integrals of local roof functions.

Findings

The bound is close to optimal in certain examples.

The result applies to systems over adelic fields satisfying the product formula.

Uses arithmetic intersection theory on toric varieties.

Abstract

We present an upper bound for the height of the isolated zeros in the torus of a system of Laurent polynomials over an adelic field satisfying the product formula. This upper bound is expressed in terms of the mixed integrals of the local roof functions associated to the chosen height function and to the system of Laurent polynomials. We also show that this bound is close to optimal in some families of examples. This result is an arithmetic analogue of the classical Bern\v{s}tein-Ku\v{s}nirenko theorem. Its proof is based on arithmetic intersection theory on toric varieties.