This thesis consists of four papers and an introduction.  In Paper I we calculate the second order derivatives of the Ronkin function of an affine polynomial in three variables. This gives an expression for the real Monge-Ampére measure associated to the hyperplane amoeba. The measure is expressed in terms of complete elliptic integrals and hypergeometric functions.  In Paper II and III we prove that a certain semi-explicit cohomological residue associated to a Cohen-Macaulay ideal or more generally an ideal of pure dimension, respectively, is annihilated precisely by the given ideal. This is a generalization of the local duality principle for the Grothendieck residue and the cohomological residue of Passare. These results follow from residue calculus, due to Andersson and Wulcan, but the point here is that our proof is more elementary. In particular, it does not rely on the desingularization theorem of Hironaka. In Paper IV we prove a global uniform Artin-Rees lemma for sections of ample line bundles over smooth projective varieties. We also prove an Artin-Rees lemma for the polynomial ring with uniform degree bounds. The proofs are based on multidimensional residue calculus. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 3: Manuscript. Paper 4. Manuscript.</p>