Geodesic
The 2-$d$ Jacobian conjecture, the $d$-inversion approximation and its natural boundary
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVII, Tome 373 (2009), pp. 210-225 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $F\in\mathbb C[X,Y]^2$ be an étale mapping of degree $\operatorname{deg}F=d$. An Étale mapping $G\in\mathbb C[X,Y]^2$ is called a $d$-inverse approximation of $F$ if $\operatorname{deg}G\le d$ and $F\circ G=(X+A(X,Y),Y+B(X,Y))$ and $G\circ F=(X+C(X,Y),Y+D(X,Y))$ where the orders of the four polynomials $A,B,C$ and $D$ are greater that $d$. It is a well known result that every $\mathbb C^2$ automorphism $F$ of degree $d$ has a $d$-inverse approximation, namely $F^{-1}$. In this paper we prove that if $F$ is a counterexample of degree $d$ to the 2-dimensional Jacobian Conjecture, then $F$ has no $d$-inverse approximation. We also give few conclusions of this result. Bibl. – 18 titles. 
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R. Peretz. The 2-$d$ Jacobian conjecture, the $d$-inversion approximation and its natural boundary. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVII, Tome 373 (2009), pp. 210-225. http://geodesic.mathdoc.fr/item/ZNSL_2009_373_a13/

[1] H. Bass, E. H. Connell, D. Wright, “The Jacobian conjecture: reduction of degree and formal expansion of the inverse”, Bull. Amer. Math. Soc., 7:2 (1982), 287–330 | DOI | MR | Zbl

[2] M. de Bondt, A. van den Essen, Nilpotent symmetric Jacobian matrices and the Jacobian Conjecture, Report No 0307, University of Nijmegen, June 2003 | MR

[3] M. de Bondt, A. van den Essen, “Nilpotent symmetric Jacobian matrices and the Jacobian conjecture. II”, J. Pure Appl. Algebra, 196 (2005), 135–148 | DOI | MR | Zbl

[4] P. L. Duren, Univalent Functions, Springer-Verlag, 1983 | MR | Zbl

[5] A. van den Essen, Polynomial Automorphisms and the Jacobian conjecture, Progress Math., 190, Birkhäuser Verlag, Boston–Basel–Berlin, 2000 | MR | Zbl

[6] Jean-Philippe Furter, “On the variety of automorphisms of the affine plane”, J. Algebra, 195 (1997), 604–623 | DOI | MR | Zbl

[7] S. Friedland, J. Milnor, “Dynamical properties of the plane polynomial automorphisms”, Ergodic Th. Dynam. Systems, 9 (1989), 67–99 | MR | Zbl

[8] H. Grauert, R. Remmert, Coherent Analytic Sheaves, Grundlehren der mathematischen Wissenschaften, 265, Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1984 | DOI | MR | Zbl

[9] H. W. E. Jung, “Über ganze birationale Transformationen der Ebene”, J. Reine Angew. Math., 184 (1942), 161–174 | DOI | MR | Zbl

[10] S. Kaliman, “Extensions of isomorphisms between affinealgebraic subvarieties of $k^n$ to automorphisms of $k^n$”, Proceedings of the AMS, 113:2 (1991), 325–334 | DOI | MR | Zbl

[11] T. Kambayashi, “Pro-affine algebras, ind-affine groups and the Jacobian problem”, J. Algebra, 185 (1996), 481–501 | DOI | MR | Zbl

[12] T. Kambayashi, “Some basic results on pro-affine algebras and ind-affine schemes”, Osaka J. Math., 40 (2003), 621–638 | MR | Zbl

[13] T. Kambayashi, “Inverse limits of polynomial rings”, Osaka J. Math., 41 (2004), 617–624 | MR | Zbl

[14] W. van der Kulk, “On polynomial rings in two variables”, Nieuw Arch. Wisk. (3), 1 (1953), 33–41 | MR | Zbl

[15] R. Peretz, “The Jacobian variety”, Acta Vietnamica (to appear)

[16] I. R. Shafarevich, Basic Algebraic Geometry, Springer-Verlag, New York–Heidelberg–Berlin, 1974 | MR | Zbl

[17] I. R. Shafarevich, “On some infinite-dimensional groups”, Rend. Mat. Appl., 25 (1966), 208–212 | MR | Zbl

[18] I. R. Shafarevich, “On some infinite-dimensional groups. II”, Math. USSR Izv., 18 (1982), 185–194 | DOI | MR | Zbl