Polynomial automorphisms and the Jacobian conjecture / Arno van den Essen.

Includes bibliographical references and indexes.

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 Grobner 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.