Ebook: Residue Currents and Bezout Identities

A very primitive form of this monograph has existed for about two and a half years in the form of handwritten notes of a course that Alain Y ger gave at the University of Maryland. The objective, all along, has been 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 [71;5]* Our original interest in the subject rested on the fact that the study of many questions in harmonic analysis, like finding all distribution solutions (or finding out whether there are any) to a system of linear partial differential equa­ tions with constant coefficients (or, more generally, convolution equations) in ]R. n, can be translated into interpolation problems in spaces of entire functions with growth conditions. This idea, which one can trace back to Euler, is the basis of Ehrenpreis's Fundamental Principle for partial differential equations [37;5], [56;5], and has been explicitly stated, for convolution equations, in the work of Berenstein and Taylor [9;5] (we refer to the survey [8;5] for complete references. ) One important point in [9;5] was the use of the Jacobi interpo­ lation formula, but otherwise, the representation of solutions obtained in that paper were not explicit because of the use of a-methods to prove interpolation results.

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The objective of this monograph is to present a coherent picture of the almost mysterious role that analytic methods and, in particular, multidimensional residue have recently played in obtaining effective estimates for problems in commutative algebra. Bezout identities, i. e., f1g1 + ... + fmgm = 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 Cn, 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.

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The objective of this monograph is to present a coherent picture of the almost mysterious role that analytic methods and, in particular, multidimensional residue have recently played in obtaining effective estimates for problems in commutative algebra. Bezout identities, i. e., f1g1 + ... + fmgm = 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 Cn, 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.
Content:
Front Matter....Pages i-xi
Residue Currents in one Dimension Different Approaches....Pages 1-20
Integral Formulas in Several Variables....Pages 21-47
Residue Currents and Analytic Continuation....Pages 49-90
The Cauchy-Weil Formula and its Consequences....Pages 91-116
Applications to Commutative Algebra and Harmonic Analysis....Pages 117-158
Back Matter....Pages 159-160

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The objective of this monograph is to present a coherent picture of the almost mysterious role that analytic methods and, in particular, multidimensional residue have recently played in obtaining effective estimates for problems in commutative algebra. Bezout identities, i. e., f1g1 + ... + fmgm = 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 Cn, 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.
Content:
Front Matter....Pages i-xi
Residue Currents in one Dimension Different Approaches....Pages 1-20
Integral Formulas in Several Variables....Pages 21-47
Residue Currents and Analytic Continuation....Pages 49-90
The Cauchy-Weil Formula and its Consequences....Pages 91-116
Applications to Commutative Algebra and Harmonic Analysis....Pages 117-158
Back Matter....Pages 159-160
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