Polynomial Automorphisms: And the Jacobian Conjecture

ISBN-13: 9783034895675 / Angielski / Miękka / 2012 / 329 str.

Arno Van Den Essen; Arno Van Den Essen

Opis

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 Grobner basis theory. A long list of ''strong'' examples and an extensive bibliography conclude the book.

Recenzje

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

--Mathematical Reviews

Spis treści

I Methods.- 1. Preliminaries.- 1.1 The formal inverse function theorem and its applications.- 1.2 Derivations.- 1.3 Locally finite derivations.- 1.4 Algorithms for locally nilpotent derivations.- 2 Derivations and polynomial automorphisms.- 2.1 Locally nilpotent derivations and polynomial automorphisms.- 2.2 Derivations and the Jacobian Condition.- 2.3 The degree of the inverse of a polynomial automorphism.- 3 Invertibility criteria and inversion formulae.- 3.1 A formula for the formal inverse.- 3.2 An invertibility algorithm for morphisms between finitely generated k-algebras.- 3.3 A resultant criterion and formula for the inversion of a polynomial map in two variables.- 4 Injective morphisms.- 4.1 Injective endomorphisms are surjective.- 4.2 Injective endomorphisms of affine algebraic sets are automorphisms.- 4.3 A short proof of theorem 4.2.1 in case V = kn and an application to the Jacobian Conjecture.- 4.4 Injective morphisms between irreducible affine varieties of the same dimension.- 5 The tame automorphism group of a polynomial ring.- 5.1 The tame automorphism group of R[X, Y].- 5.2 The tame automorphism group in dimension ? 3.- 5.3 Embeddings of affine algebraic varieties and tame automorphisms.- 5.4 The Abhyankar-Moh theorem.- 6 Stabilization Methods.- 6.1 The stabilization principle: some instructive examples.- 6.2 Stable equivalence.- 6.3 Applications to the Jacobian Conjecture.- 6.4 Gorni-Zampieri pairing.- 7 Polynomial maps with nilpotent Jacobian.- 7.1 Hubbers’ theorem and a dependence problem.- 7.2 The class H (n, A).- 7.3 H(n, A), D(n, A) and stable tameness.- 7.4 Strongly nilpotent Jacobian matrices.- II Applications.- 8 Applications of polynomial mappings to dynamical systems.- 8.1 The Markus-Yamabe Conjecture and a problem of LaSalle; some background.- 8.2 The Markus-Yamabe Conjecture and the LaSalle problem in dimension two.- 8.3 The story of the solution of the Markus-Yamabe Conjecture.- 8.4 Meisters’ cubic-linear linearization conjecture and the MYC revisited.- 9 Group actions.- 9.1 Algebraic group actions: an introduction.- 9.2 Hilbert’s finiteness theorem.- 9.3 Constructive invariant theory: Derksen’s algorithm to compute the invariants for reductive groups.- 9.4 A linearization conjecture for reductive group actions.- 9.5 Ga-actions.- 9.6 Ga-actions and Hilbert’s fourteenth problem.- 10 The Jacobian Conjecture.- 10.1 Injectivity and invertibility of differentiable maps and the real Jacobian Conjecture.- 10.2 The two-dimensional Jacobian Conjecture.- 10.3 Polynomial maps with integer coefficients and the Jacobian Conjecture in positive characteristic.- 10.4 D-modules and the Jacobian Conjecture.- 10.5 Endomorphisms sending coordinates to coordinates.- 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.