## A bound for $[\mathbb{C}(x,y,z):\mathbb{C}(p,q,r)]$, where $\operatorname{Jac}(p,q,r) \in \mathbb{C}^{\times}$

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 $$\leq (\min{ \{\deg(p),\deg(q),\deg(r)\})^2}$$.

Any hints and comments are welcome! Thank you.