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A+ CATEGORY SCIENTIFIC UNIT

The Jacobian Conjecture: symmetric reduction and solution in the symmetric cubic linear case

Volume 87 / 2005

Ludwik M. Drużkowski Annales Polonici Mathematici 87 (2005), 83-92 MSC: 14R15, 14R10. DOI: 10.4064/ap87-0-7

Abstract

Let denote \mathbb R or \mathbb C, n>1. The Jacobian Conjecture can be formulated as follows: If F:{\mathbb K}^n \to {\mathbb K}^n is a polynomial map with a constant nonzero jacobian, then F is a polynomial automorphism. Although the Jacobian Conjecture is still unsolved even in the case n=2, it is convenient to consider the so-called Generalized Jacobian Conjecture (for short (GJC)): the Jacobian Conjecture holds for every n>1. We present the reduction of (GJC) to the case of F of degree 3 and of symmetric homogeneous form and prove (JC) for maps having cubic linear form with symmetric F'(x), more precisely: polynomial maps of cubic linear form with symmetric F'(x) and constant nonzero jacobian are tame automorphisms.

Authors

  • Ludwik M. DrużkowskiInstitute of Mathematics
    Jagiellonian University
    Reymonta 4
    30-059 Kraków, Poland
    e-mail

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