IJPAM: Volume 80, No. 4 (2012)


Kristen Lampe
Department of Mathematics
Carroll University
100, North East Avenue, Waukesha, Wisconsin, 53186, USA

Abstract. The Jacobian conjecture deals with a polynomial map $F:k^n \to
k^n$ where $k$ is a field of characteristic zero. It says that if the Jacobian determinant is a unit in $k$, then $F$ has a polynomial inverse. A known equivalent statement of this conjecture uses the vanishing of a class of polynomials $P_r$, related to $F$.

A known formula expresses the inverse of $F$ as a power series whose coefficients are parameterized by labeled, rooted forests. Here, we develop a new expression for the polynomials $P_r$, in terms of labeled, rooted trees. We then express the coefficients of $P_r$ in terms of the coefficients of $F$. Using a known counting method originally developed to express the inverse of $F$ in terms of labeled, rooted forests, we explore the relationship between the coefficients of the $P_r$ and the coefficients of the inverse of $F$.

The result is a simplification of the inverse formula for $F$ 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
Year: 2012
Volume: 80
Issue: 4