Let \( F \in \mathbb{C}{\left[ {X,\,Y} \right]^2} \) be an étale map of degree deg F = d. An étale map \( G \in \mathbb{C}{\left[ {X,Y} \right]^2} \) is called a d-inverse approximation of F if deg G ≤ d and F ◦ G =(X + A(X, Y), Y + B(X, Y)) and G ◦ F =(X + C(X, Y), Y + D(X, Y)), where the orders of the four polynomials A, B, C, and D are greater than 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 two-dimensional Jacobian conjecture, then F has no d-inverse approximation. We also give few consequences of this result. Bibliography: 18 titles.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 373, 2009, pp. 210–225.
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Peretz, R. The Jacobian conjecture, the d-inversion approximation and its natural boundary. J Math Sci 168, 428–436 (2010). https://doi.org/10.1007/s10958-010-9995-9
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DOI: https://doi.org/10.1007/s10958-010-9995-9