Some Results On The Jacobian Conjecture And Polynomial Automorphisms

Abstract

In this paper, we will first show that, the homogeneous polynomials which satisfy the Jacobian condition are injective on the lines that pass through the origin. Secondly, we will show that $F$ and $G^{\prime }$ are paired, where $F$ is a Druzkowski map and $G^{\prime }$ is a cubic homogeneous polynomial which related to $F$. Finally, we will find a more exactly bound for the degree of $F^{-1}$, where $F$ is a invertible map.

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