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Register a new userThe classification of some polynomial maps with nilpotent Jacobians
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In the paper, we first classify all polynomial maps $H$ of the following form: $H=big(H_1(x_1,x_2,ldots,x_n),H_2(x_1,x_2),H_3(x_1,x_2),ldots,H_n(x_1,x_2)big)$ with $JH$ nilpotent. After that, we generalize the structure of $H$ to $H=big(H_1(x_1,x_2,ldots,x_n),H_2(x_1,x_2),H_3(x_1,x_2,H_1),ldots,H_n(x_1,x_2,H_1)big)$.
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Let $K$ be any field with $textup{char}K eq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ with $textup{rk} JHleq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map then $F$ is invertible, and furthermore $F$ is tame if the dimension $n eq 4$.
This article has been withdrown by the author.
In this paper we study principally polarized abelian varieties that admit an automorphism of prime order $p>2$. It turns out that certain natural conditions on the multiplicities of its action on the differentials of the first kind do guarantee that those polarized varieties are not jacobians of curves.
We classify all quadratic homogeneous polynomial maps $H$ and Keller maps of the form $x + H$, for which $rk J H = 3$, over a field $K$ of arbitrary characteristic. In particular, we show that such a Keller map (up to a square part if $char K=2$) is a tame automorphism.
This paper has been withdrawn by the author due to a crucial argument error at p.10.
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