# On the density of S-adic integers near some projective G-varieties

## TL;DR

This paper establishes conditions under which systems of inequalities with homogeneous polynomials over S-adic fields have nontrivial S-integral solutions, using adelic geometry and strong approximation, with applications to quadratic and linear forms.

## Contribution

It introduces general criteria for the existence of S-integral solutions to polynomial inequalities, leveraging adelic geometry and strong approximation techniques.

## Key findings

- Conditions for nontrivial S-integral solutions are provided.
- Applications demonstrated for quadratic and linear forms.
- The approach combines adelic geometry with strong approximation.

## Abstract

We provide some general conditions which ensure that a system of inequalities involving homogeneous polynomials with coefficients in a S-adic field has nontrivial S-integral solutions. The proofs are based on the strong approximation property for Zariski-dense subgroups and adelic geometry of numbers. We give two examples of applications for systems involving quadratic and linear forms.