# Moscow Mathematical Journal

Volume 21, Issue 1, January–March 2021  pp. 129–173.

Bounds for Multivariate Residues and for the Polynomials in the Elimination Theorem

Authors:  Martín Sombra (1) and Alain Yger (2)
Author institution:(1) Institució Catalana de Recerca i Estudis Avançats (ICREA). Passeig Lluís Companys 23, 08010 Barcelona, Spain
Departament de Matemàiques i Informàtica, Universitat de Barcelona. Gran Via 585, 08007 Barcelona, Spain
(2) Institut de Mathématiques, Université de Bordeaux. 351 cours de la Libération, 33405 Talence, France

Summary:

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.

2010 Math. Subj. Class. Primary: 32A27; Secondary: 11G50, 14Q20.

Keywords: Residues, membership problems, height.