Let $p=p(x,y), q=q(x,y) \in \mathbb{C}[x,y]$ be a Jacobian pair, namely, $Jac(p,q)=p_xq_y-p_yq_x \in \mathbb{C}^{\times}$. Denote the total degrees of $p$ and $q$ by $\deg(p)$ and $\deg(q)$, and denote the degree of the (finite) field extension $\mathbb{C}(p,q) \subseteq \mathbb{C}(x,y)$ by $d$.

A nice result due to P. I. Katsylo and Y. Zhang says that $d \leq \min\{\deg(p),\deg(q)\}$; see Katsylo's paper, Theorem 1 and Zhang's thesis, main result.

My question:

Is it possible to generalize this result to higher dimensions?

More elaborately:

Let $f_1,\ldots,f_n \in \mathbb{C}[x_1,\ldots,x_n]$, $n \geq 2$, with $Jac(f_1,\ldots,f_n) \in \mathbb{C}^{\times}$, and denote the degree of the (finite) field extension $\mathbb{C}(f_1,\ldots,f_n) \subseteq \mathbb{C}(x_1,\ldots,x_n)$ by $d_n$. (So $d_2=d$).

Fix $n \geq 3$. Does one of the following inequalities hold?

(1) $d_n \leq \min\{\deg(f_1),\ldots,\deg(f_n)\}$.

(2) $d_n \leq (\min\{\deg(f_1),\ldots,\deg(f_n)\})^{n-1}$.


(i) If I am not wrong, we have the following claim: If inequality (2) holds for all $n$, then the generalized Jacobian Conjecture is true. An explanation for this claim can be found in Proposition 3.5; it is based on the result that is mentioned here.

(ii) Unfortunately, it seems that a proof for (1) or (2) should be very difficult (at least to me; hopefully not to someone else).

Thank you very much! Any comments are welcome!


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