Berenstein, Carlos A.Yger, A.We provide new tools to compute multidimensional residues for rational functions, even over fields of positive characteristic. As a corollary one obtains solutions of the Betout equation for polynomials over a ring with a site that have almost optimal estimates for degree and size.en-USalgorithmscomputational complexitycomputational geometryoptimal controlsystem theoryIntelligent Control SystemsTechnical Report