Y. Zhang (in his PhD thesis) and P. I. Katsylo proved the following nice result; the two proofs are different, see: Zhang's thesis and Katsylo's paper:

Let $f: (x,y) \mapsto (p,q)$ be a $k$-algebra endomorphism of $\mathbb{C}[x,y]$ having an invertible Jacobian, namely, $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$. Then the degree of the field extension $\mathbb{C}(p,q) \subseteq \mathbb{C}(x,y)$ is $\leq \min{ \{\deg(p),\deg(q)\}}$.

Now let $f: (x,y,z) \mapsto (p,q,r)$ be a $k$-algebra endomorphism of $\mathbb{C}[x,y,z]$ having an invertible Jacobian, namely, $\operatorname{Jac}(p,q,r) \in \mathbb{C}-\{0\}$.

Is the following claim true?

The degree of the field extension $\mathbb{C}(p,q,r) \subseteq \mathbb{C}(x,y,z)$ is $\leq (\min{ \{\deg(p),\deg(q),\deg(r)\})^2}$.

Remark: I have tried to generalize Zhang's proof to higher dimensions (or at least to dimension three), but I have a few gaps (I have tried to mimick his proof). I do not know how to generalize Katsylo's proof to dimension three.

Any hints and comments are welcome! Thank you.

  • 3
    $\begingroup$ If there exists an endomorphism of $\mathbb{C}[x,y]$ with Jacobian invertible, and degree of the field extension greater than one (I know this contradicts Jacobian conjecture), then take an endomorphism of $\mathbb{C}[x,y,z]$ by using the one for $x,y$ and sending $z$ to itself. Clearly, the Jacobian condition is met and the degree of the extension is still greater than one, which is larger than the degree of $z$. $\endgroup$ – Mohan Jun 13 '19 at 18:01
  • $\begingroup$ This paper arxiv.org/abs/1610.01621 says that if one has a bound (a la Gabber) with min degree at power dimension minus one then the Jcobian Conjecture would follow. $\endgroup$ – Abdelmalek Abdesselam Jun 13 '19 at 20:16
  • $\begingroup$ Thank you, Mohan and Abdelmalek Abdesselam. Yes, I know that if my above three-dimensional claim is true, then the two-dimensional Jacobian Conjecture is ture. Truly, this is why I have posted this question, hoping that someone more familiar than me with the notions in Zhang's proof and in Katsylo's proof will be able to either find a proof for the claim (and let us know that it is provable) or comment that it seems difficult to generalize their proofs to dimension three. $\endgroup$ – user237522 Jun 14 '19 at 5:26
  • $\begingroup$ In arxiv.org/pdf/1006.5801v1.pdf A. van den Essen says: "...Therefore I often stated in public the following dictum: If you have a conjecture which implies the (two-dimensional?) Jacobian Conjecture, but is not equivalent to it, then you can be sure that your conjecture is false". Therefore, if his statement is true, then probably my three-dimensional claim is false. $\endgroup$ – user237522 Jun 14 '19 at 5:39

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