# Is it possible to generalize a result of Katsylo-Zhang concerning the two-dimensional JC?

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

Remarks:

(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!