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Polynomial Automorphisms

and the Jacobian Conjecture

Arno van den Essen (Autor)

Buch | Hardcover

XVIII, 329 Seiten

2000 | 2000
Springer Basel (Verlag)
978-3-7643-6350-5 (ISBN)

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Motivated by some notorious open problems, such as the Jacobian conjecture and the tame generators problem, the subject of polynomial automorphisms has become a rapidly growing field of interest. This book, the first in the field, collects many of the results scattered throughout the literature. It introduces the reader to a fascinating subject and brings him to the forefront of research in this area. Some of the topics treated are invertibility criteria, face polynomials, the tame generators problem, the cancellation problem, exotic spaces, DNA for polynomial automorphisms, the Abhyankar-Moh theorem, stabilization methods, dynamical systems, the Markus-Yamabe conjecture, group actions, Hilbert's 14th problem, various linearization problems and the Jacobian conjecture. The work is essentially self-contained and aimed at the level of beginning graduate students. Exercises are included at the end of each section. At the end of the book there are appendices to cover used material from algebra, algebraic geometry, D-modules and Gröbner basis theory. A long list of ''strong'' examples and an extensive bibliography conclude the book.

I Methods.- 1. Preliminaries.- 2 Derivations and polynomial automorphisms.- 3 Invertibility criteria and inversion formulae.- 4 Injective morphisms.- 5 The tame automorphism group of a polynomial ring.- 6 Stabilization Methods.- 7 Polynomial maps with nilpotent Jacobian.- II Applications.- 8 Applications of polynomial mappings to dynamical systems.- 9 Group actions.- 10 The Jacobian Conjecture.- III Appendices.- A Some commutative algebra.- A.1 Rings.- A.2 Modules.- A.3 Localization.- A.4 Completions.- A.5 Finiteness conditions and integral extensions.- A.6 The universal coefficients method.- B Some basic results from algebraic geometry.- B.1 Algebraic sets.- B.2 Morphisms of irreducible affine algebraic varieties.- C Some results from Gröbner basis theory.- C.1 Definitions and basic properties.- C.2 Applications: several algorithms.- D Flatness.- D.1 Flat modules and algebras.- D.2 Flat morphisms between affine algebraic varieties.- E.2 Direct and inverse images.- F Special examples and counterexamples.- Authors Index.

"...This book is a valuable reference for the study of polynomial automorphisms, due to its breadth of coverage and clarity of exposition."

--Mathematical Reviews

Erscheint lt. Verlag	1.9.2000
Reihe/Serie	Progress in Mathematics
Zusatzinfo	XVIII, 329 p.
Verlagsort	Basel
Sprache	englisch
Maße	155 x 235 mm
Gewicht	720 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Schlagworte	Algebra • Algebraic Geometry • algebraic group • Algebraic Varieties • automorphism • Commutative algebra • Dimension • Field • Grad • Gröbner basis • Invariant theory • matrices
ISBN-10	3-7643-6350-9 / 3764363509
ISBN-13	978-3-7643-6350-5 / 9783764363505
Zustand	Neuware