Multidimensional residues and complexity problems

Our main objective in this note is to show how multidimensional residues can be used to make explicit division formulas in polynomial algebras; we will also discuss division problems with Laurent polynomials and detail, in some particular cases, the methods that can be used to compute the complete sum of residues of a rational function; we will finally indicate the relation between the search of effective bounds and the numerical computation of multidimensional integrals.