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

We present several upper bounds for the height of global residues of rational forms on an affine variety defined over $\mathbb{Q}$. As an application, 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 defined over $\mathbb{Q}$. This is an arithmetic analogue of Jelonek's effective elimination theorem, and it plays a crucial role in the proof of our bounds for the height of global residues.