  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.
