IJPAM: Volume 80, No. 4 (2012)
TO THE JACOBIAN CONJECTURE
Department of Mathematics
100, North East Avenue, Waukesha, Wisconsin, 53186, USA
Abstract. The Jacobian conjecture deals with a polynomial map where is a field of characteristic zero. It says that if the Jacobian determinant is a unit in , then has a polynomial inverse. A known equivalent statement of this conjecture uses the vanishing of a class of polynomials , related to .
A known formula expresses the inverse of as a power series whose coefficients are parameterized by labeled, rooted forests. Here, we develop a new expression for the polynomials , in terms of labeled, rooted trees. We then express the coefficients of in terms of the coefficients of . Using a known counting method originally developed to express the inverse of in terms of labeled, rooted forests, we explore the relationship between the coefficients of the and the coefficients of the inverse of .
The result is a simplification of the inverse formula for
under the Jacobian hypothesis.
Received: June 19, 2012
AMS Subject Classification: 05E40, 12Y05
Key Words and Phrases: Jacobian conjecture, counting formula, polynomials, reversion, trees, forests
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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395