Residue Currents and Bezout Identities

1. Residue Currents in one Dimension. Different Approaches.- 1. Residue attached to a holomorphic function.- 2. Some other approaches to the residue current.- 3. Some variants of the classical Pompeiu formula.- 4. Some applications of Pompeiu's formulas. Local results.- 5. Some applications of Pompeiu's formulas. Global results.- References for Chapter 1.- 2. Integral Formulas in Several Variables.- 1. Chains and cochains, homology and cohomology.- 2. Cauchy's formula for test functions.- 3. Weighted Bochner-Martinelli formulas.- 4. Weighted Andreotti-Norguet formulas.- 5. Applications to systems of algebraic equations.- References for Chapter 2.- 3. Residue Currents and Analytic Continuation.- 1. Leray iterated residues.- 2. Multiplication of principal values and residue currents.- 3. The Dolbeault complex and the Grothendieck residue.- 4. Residue currents.- 5. The local duality theorem.- References for Chapter 3.- 4. The Cauchy-Weil Formula and its Consequences.- 1. The Cauchy-Weil formula.- 2. The Grothendieck residue in the discrete case.- 3. The Grothendieck residue in the algebraic case.- References for Chapter 4.- 5. Applications to Commutative Algebra and Harmonic Analysis.- 1. An analytic proof of the algebraic Nullstellensatz.- 2. The membership problem.- 3. The Fundamental Principle of L. Ehrenpreis.- 4. The role of the Mellin transform.- References for Chapter 5.