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Residue currents and Bezout identities. (English) Zbl 0802.32001

Progress in Mathematics (Boston, Mass.). 114. Basel: Birkhäuser. xi, 158 p. (1993).
The author’s abstract: “The objective of this monograph is to present a coherent picture of the almost mysterious role that analytic methods and, in particular, multidimensional residues have recently played in obtaining effective estimates for problems in commutative algebra.
Bezout identities, i.e., \(f_ 1 g_ 1+\cdots+ f_ m g_ m= 1\), appear naturally in many problems, for example in commutative algebra in the Nullstellensatz, and in signal processing in the deconvolution problem. One way to solve them is by using explicit interpolation formulas in \(\mathbb{C}^ n\), and these depend on the theory of multidimensional residues. The authors present this theory in detail, in a form developed by them, and illustrate its applications to the effective Nullstellensatz and to the fundamental principle for convolution equations”.

MSC:

32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32B05 Analytic algebras and generalizations, preparation theorems
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
32A27 Residues for several complex variables
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