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 Zap. Nauchn. Sem. POMI, 2009, Volume 373, Pages 210–225 (Mi znsl3584)

The 2-$d$ Jacobian conjecture, the $d$-inversion approximation and its natural boundary

R. Peretz

Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva, Israel

Abstract: 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.

Key words and phrases: the Jacobian conjecture, étale morphisms, inversion formulas, polynomial automorphisms, natural boundary.

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English version:
Journal of Mathematical Sciences (New York), 2010, 168:3, 428–436

Document Type: Article
UDC: 517.55+512.71
Language: English

Citation: R. Peretz, “The 2-$d$ Jacobian conjecture, the $d$-inversion approximation and its natural boundary”, Representation theory, dynamical systems, combinatorial methods. Part XVII, Zap. Nauchn. Sem. POMI, 373, POMI, St. Petersburg, 2009, 210–225; J. Math. Sci. (N. Y.), 168:3 (2010), 428–436

Citation in format AMSBIB
\Bibitem{Per09} \by R.~Peretz \paper The 2-$d$ Jacobian conjecture, the $d$-inversion approximation and its natural boundary \inbook Representation theory, dynamical systems, combinatorial methods. Part~XVII \serial Zap. Nauchn. Sem. POMI \yr 2009 \vol 373 \pages 210--225 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl3584} \transl \jour J. Math. Sci. (N. Y.) \yr 2010 \vol 168 \issue 3 \pages 428--436 \crossref{https://doi.org/10.1007/s10958-010-9995-9} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77954757959}