# Questions tagged [jacobian-conjecture]

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### Injectivity of Keller maps

Let $M: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$, $(x,y) \mapsto (p,q)$, with $p,q \in \mathbb{C}[x,y]$ satisfying $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$. Such a polynomial map is ...
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### Concerning a result of E. Formanek

A result of E. Formanek, in its two-dimensional version, says: Let $k$ be a field of characteristic zero and let $R=k[x,y]$ be the polynomial ring in two variables. Let $p,q \in R$ have invertible ...
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### A Jacobian pair $(p,q)$ such that $\gcd(\deg(p),\deg(q))=2P$, $P \geq 5$ is prime

Let $p=p(x,y),q=q(x,y) \in \mathbb{C}[x,y]$ be a Jacobian pair, namely, $p_xq_y-p_yq_x \in \mathbb{C}^*$. Denote $a:= \deg(p)$ and $b:= \deg(q)$, where $\deg()$ denotes the total degree ($(1,1)$-...
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### Reference request concerning the generalized Jacobian Conjecture

On page 287, A. van den Essen says: Furthermore one can show that it suffices to prove JC for all $n \geq 2$ and for all $F$'s of the form: $F=(l_1,\ldots,l_r,x_{r+1}+M_{r+1},\ldots,x_n+M_n)$ ...
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### Rectangular Newton polygon of a Jacobian pair

Let $p,q \in k[x,y]$, $k$ is a field of characteristic zero. By definition, $p,q$ is a Jacobian pair if their Jacobian is invertible in $k[x,y]$, namely, $p_xq_y-p_yq_x \in k^*$, and $p,q$ is an ...
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### Does the Polarization Theorem for $A_1(k)$ has an analogue for $k[x,y]$?

There is an interesting theorem about the first Weyl algebra $A_1(k)= k \langle x,y | yx-xy= 1 \rangle$, $k$ is a field of characteristic zero, the Polarization Theorem, Corollary 5.5 by A. Joseph. ...