Alekos Vidras

Residue Currents and Bezout Identities

Introduction 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 Pompeius formulas. Local results 5. Some applications of Pompeius formulas. Global results References for Chapter 1 2. Integral formulas in several variables 1. Chains and cochains, homology and cohomology 2. Cauchys 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.

On some generalizations of Jacobi's residue formula

Abstract Using Bochner–Martinelli type residual currents we prove some generalizations of Jacobis residue formula, which allow proper polynomial maps to have ‘common zeroes at infinity’, in projective or toric situations.

On asymptotic approximations of the residual currents

We use a D-module approach to discuss positive examples for the existence of the unrestricted limit of the integrals involved in the approximation to the Coleff-Herrera residual currents (in the complete intersection case.) Our results provide also asymptotic developments for these integrals. ∗AMS classification number: 32A27, 32C30.

Coleff–Herrera Currents Revisited

In the present paper, we describe the recent approach to residue currents by Andersson, Bjork, and Samuelsson (Andersson in Ann. Fac. Sci. Toulouse Math. Ser. 18(4):651–661, 2009; Bjork in The Legacy of Niels Henrik Abel, pp. 605–651, Springer, Berlin, 2004; Bjork and Samuelsson in J. Reine Angew. Math. 649:33–54, 2010), focusing primarily on the methods inspired by analytic continuation (that were initiated in a quite primitive form in Berenstein et al. in Residue Currents and Bezout Identities. Progress in Mathematics, vol. 114, Birkhauser, Basel, 1993). Coleff–Herrera currents (with or without poles) play indeed a crucial role in Lelong–Poincare-type factorization formulas for integration currents on reduced closed analytic sets. As revealed by local structure theorems (which can also be understood as global when working on a complete algebraic manifold due to the GAGA principle), such objects are of algebraic nature (antiholomorphic coordinates playing basically the role of “inert” constants). Thinking about division or duality problems instead of intersection ones (especially in the “improper” setting, which is certainly the most interesting), it happens then to be necessary to revisit from this point of view the multiplicative inductive procedure initiated by Coleff and Herrera (Lecture Notes in Mathematics, vol. 633, Springer, Berlin, 1978), this being the main objective of this presentation. In homage to the pioneering work of Leon Ehrenpreis, to whom we are both deeply indebted, and as a tribute to him, we also suggest a currential approach to the so-called Nœtherian operators that remain the key stone in various formulations of Leon’s Fundamental Principle.

On Small Complete Sets of Functions

Using Local Residues and the Duality Principle a multidimensional variation of the completeness theorems by T. Carleman and A. F. Leontiev is proven for the space of holomorphic functions defined on a suitable open strip T� ⊂ C2. The completeness theorem is a direct consequence of the Cauchy Residue Theorem in a torus. With suitable modifications the same result holds in C n .

Local residues and discrete sets of uniqueness

If then by T Гς , we denote the domain of C 2 defined by . If B(T Гσ denotes the space of bounded holomorphic functions defined in the domain T Гς then the following is true. Let be a sequence of points tending to infinity and satisfying . Let also the sequence satisfies for every and for some constant 0 < d < l. Assume that the counting function n{λ m }(r) satisfies eventually. If then the sequence {(λ n , μ n )} is a set of uniqueness for the space B(T Гς ). That is if g e B(T Гς ) and g(λ n , μ n )=0 for every ne N then g≡0. We prove the above statement by using local residues an of a global meromorphic (2,0) form, whose isolated “poles” are the points (λ n , μ n ) n e N, in order to express the inverse Laplace Transform σ of an appropriately chosen meromorphic form as a “series” . As a consequence of this result we have the following variation of a Muntz–Szasz theorem in Let be a sequence as in the uniqueness theorem above. Then the linear span of is dense in the space of continuous functions vanishin...