TY - JOUR

T1 - The Jacobian conjecture, the d-inversion approximation and its natural boundary

AU - Peretz, R.

PY - 2010/7/12

Y1 - 2010/7/12

N2 - Let F ∈ ℂ[X,Y]2 be an étale map of degree deg F = d. An étale map G ∈ ℂ[X,Y]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 ℂ-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.

AB - Let F ∈ ℂ[X,Y]2 be an étale map of degree deg F = d. An étale map G ∈ ℂ[X,Y]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 ℂ-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.

UR - http://www.scopus.com/inward/record.url?scp=77954757959&partnerID=8YFLogxK

U2 - 10.1007/s10958-010-9995-9

DO - 10.1007/s10958-010-9995-9

M3 - Article

AN - SCOPUS:77954757959

VL - 168

SP - 428

EP - 436

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 3

ER -