# Bounds for multivariate residues and for the polynomials in the elimination theorem

## TL;DR

This paper establishes upper bounds for the height of global residues and the polynomials involved in the elimination theorem on affine varieties, providing an arithmetic analogue to Jelonek's effective elimination theorem.

## Contribution

It introduces new upper bounds for the height of global residues and the polynomials in the elimination theorem, extending Jelonek's theorem arithmetically.

## Key findings

- Upper bounds for the height of global residues
- Upper bounds for the coefficients in the Bergman-Weil trace formula
- Upper bounds for degree and height of elimination polynomials

## Abstract

We present several upper bounds for the height of global residues of rational forms on an affine variety. As a consequence, we deduce upper bounds for the height of the coefficients in the Bergman-Weil trace formula. We also present upper bounds for the degree and the height of the polynomials in the elimination theorem on an affine variety. This is an arithmetic analogue of Jelonek's effective elimination theorem, that plays a crucial role in the proof of our bounds for the height of global residues.