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Register a new userQuadratic homogeneous polynomial maps $H$ and Keller maps $x+H$ with $rk JH=3$
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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.
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We compute by hand all quadratic homogeneous polynomial maps $H$ and all Keller maps of the form $x + H$, for which ${rm rk} J H = 3$, over a field of arbitrary characteristic. Furthermore, we use computer support to compute Keller maps of the form $x + H$ with ${rm rk} J H = 4$, namely: $bullet$ all such maps in dimension $5$ over fields with $frac12$; $bullet$ all such maps in dimension $6$ over fields without $frac12$. We use these results to prove the following over fields of arbitrary characteristic: for Keller maps $x + H$ for which ${rm rk} J H le 4$, the rows of $J H$ are dependent over the base field.
We classify Keller maps $x + H$ in dimension $n$ over fields with $tfrac16$, for which $H$ is homogeneous, and (1) deg $H = 3$ and rk $JH le 2$; (2) deg $H = 3$ and $n le 4$; (3) deg $H = 4$ and $n le 3$; (4) deg $H = 4 = n$ and $H_1, H_2, H_3, H_4$ are linearly dependent over $K$. In our proof of these classifications, we formulate (and prove) several results which are more general than needed for these classifications. One of these results is the classification of all homogeneous polynomial maps $H$ as in (1) over fields with $tfrac16$.
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 $n eq 4$.
Let $K$ be any field and $x = (x_1,x_2,ldots,x_n)$. We classify all matrices $M in {rm Mat}_{m,n}(K[x])$ whose entries are polynomials of degree at most 1, for which ${rm rk} M le 2$. As a special case, we describe all such matrices $M$, which are the Jacobian matrix $J H$ (the matrix of partial derivatives) of a polynomial map $H$ from $K^n$ to $K^m$. Among other things, we show that up to composition with linear maps over $K$, $M = J H$ has only two nonzero columns or only three nonzero rows in this case. In addition, we show that ${rm trdeg}_K K(H) = {rm rk} J H$ for quadratic polynomial maps $H$ over $K$ such that $frac12 in K$ and ${rm rk} J H le 2$. Furthermore, we prove that up to conjugation with linear maps over $K$, nilpotent Jacobian matrices $N$ of quadratic polynomial maps, for which ${rm rk} N le 2$, are triangular (with zeroes on the diagonal), regardless of the characteristic of $K$. This generalizes several results by others. In addition, we prove the same result for Jacobian matrices $N$ of quadratic polynomial maps, for which $N^2 = 0$. This generalizes a result by others, namely the case where $frac12 in K$ and $N(0) = 0$.
Motivated by the dynamical uniform boundedness conjecture of Morton and Silverman, specifically in the case of quadratic polynomials, we give a formal construction of a certain class of dynamical analogues of classical modular curves. The preperiodic points for a quadratic polynomial map may be endowed with the structure of a directed graph satisfying certain strict conditions; we call such a graph admissible. Given an admissible graph $G$, we construct a curve $X_1(G)$ whose points parametrize quadratic polynomial maps -- which, up to equivalence, form a one-parameter family -- together with a collection of marked preperiodic points that form a graph isomorphic to $G$. Building on work of Bousch and Morton, we show that these curves are irreducible in characteristic zero, and we give an application of irreducibility in the setting of number fields. We end with a discussion of the Galois theory associated to the preperiodic points of quadratic polynomials, including a certain Galois representation that arises naturally in this setting.
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