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Softcover ISBN:  9781470471125 
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Softcover ISBN:  9781470471125 
Product Code:  SURV/275 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470474898 
Product Code:  SURV/275.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470471125 
eBook ISBN:  9781470474898 
Product Code:  SURV/275.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 275; 2023; 533 ppMSC: Primary 13; 14; 32; 42
Residue theory is an active area of complex analysis with connections and applications to fields as diverse as partial differential and integral equations, computer algebra, arithmetic or diophantine geometry, and mathematical physics. Multidimensional Residue Theory and Applications defines and studies multidimensional residues via analytic continuation for holomorphic bundlevalued current maps. This point of view offers versatility and flexibility to the tools and constructions proposed, allowing these residues to be defined and studied outside the classical case of complete intersection. The book goes on to show how these residues are algebraic in nature, and how they relate and apply to a wide range of situations, most notably to membership problems, such as the Briançon–Skoda theorem and Hilbert's Nullstellensatz, to arithmetic intersection theory and to tropical geometry.
This book will supersede the existing literature in this area, which dates back more than three decades. It will be appreciated by mathematicians and graduate students in multivariate complex analysis. But thanks to the gentle treatment of the onedimensional case in Chapter 1 and the rich background material in the appendices, it may also be read by specialists in arithmetic, diophantine, or tropical geometry, as well as in mathematical physics or computer algebra.
ReadershipGraduate students and researchers interested in residue theory.

Table of Contents

Chapters

Residue calculus in one variable

Residue currents: A multiplicative approach

Residue currents: A bundle approach

Bochner–Martinelli kernels and weights

Integral closure, Briançon–Skoda type theorems

Residue calculus and trace formulae

Miscellaneous applications: Intersection, division

Complex manifolds and analytic spaces

Holomorphic bundles over complex analytic spaces

Positivity on complex analytic spaces

Various concepts in algebraic or analytic geometry


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Residue theory is an active area of complex analysis with connections and applications to fields as diverse as partial differential and integral equations, computer algebra, arithmetic or diophantine geometry, and mathematical physics. Multidimensional Residue Theory and Applications defines and studies multidimensional residues via analytic continuation for holomorphic bundlevalued current maps. This point of view offers versatility and flexibility to the tools and constructions proposed, allowing these residues to be defined and studied outside the classical case of complete intersection. The book goes on to show how these residues are algebraic in nature, and how they relate and apply to a wide range of situations, most notably to membership problems, such as the Briançon–Skoda theorem and Hilbert's Nullstellensatz, to arithmetic intersection theory and to tropical geometry.
This book will supersede the existing literature in this area, which dates back more than three decades. It will be appreciated by mathematicians and graduate students in multivariate complex analysis. But thanks to the gentle treatment of the onedimensional case in Chapter 1 and the rich background material in the appendices, it may also be read by specialists in arithmetic, diophantine, or tropical geometry, as well as in mathematical physics or computer algebra.
Graduate students and researchers interested in residue theory.

Chapters

Residue calculus in one variable

Residue currents: A multiplicative approach

Residue currents: A bundle approach

Bochner–Martinelli kernels and weights

Integral closure, Briançon–Skoda type theorems

Residue calculus and trace formulae

Miscellaneous applications: Intersection, division

Complex manifolds and analytic spaces

Holomorphic bundles over complex analytic spaces

Positivity on complex analytic spaces

Various concepts in algebraic or analytic geometry