# Is it true that $[k(x,y):k(f,g)] \leq \deg_{1,1}(f) \deg_{1,1}(g)$? $f,g \in k[x,y]$

Let $$f,g \in k[x,y]$$, $$k$$ is a field of characteristic zero.

Denote $$d_1=\deg_{1,0}(f)=\deg_x(f)$$, $$d_2=\deg_{0,1}(f)=\deg_y(f)$$,$$d=\deg_{1,1}(f)$$

and

$$e_1=\deg_{1,0}(f)=\deg_x(g)$$, $$e_2=\deg_{0,1}(f)=\deg_y(g)$$, $$e=\deg_{1,1}(g)$$.

It is known that $$[k(x,y):k(x,f)]=d_2$$, $$[k(x,y):k(y,f)]=d_1$$, $$[k(x,y):k(x,g)]=e_2$$, $$[k(x,y):k(y,f)]=e_1$$; see the first comment to this post.

Also, in the thesis of Y. Zhang, he proved that a Jacobian pair $$Jac(f,g)=f_xg_y-f_yg_x \in k^*$$ (here I am not assuming that $$f,g$$ is a Jacobian pair!) satisfy: $$[k(x,y):k(f,g)] \leq \min \{\deg_y(f),\deg_y(g)\}$$; this result does not hold in general, for example, $$[k(x,y):k(x,y^{10})]=10 > \min \{0,10\}$$, even if we take total degrees: $$[k(x,y):k(x,y^{10})]=10 > \min \{1,10\}$$.

Question: Is $$[k(x,y):k(f,g)] \leq \deg_{1,1}(f) \deg_{1,1}(g)$$ true?

I think that the claim is true, and there is a result concerning zeros of polynomials which may be relevant here, but I am not sure.

An attempt to find a solution: $$k(x,f) \subseteq k(x,f,g) \subseteq k(x,y)$$, hence $$[k(x,y): k(x,f,g)]| d_2$$, and similarly, $$k(x,g) \subseteq k(x,f,g) \subseteq k(x,y)$$, hence $$[k(x,y): k(x,f,g)] | e_2$$.

(Similarly, $$[k(x,y): k(y,f,g)] | d_1, e_1$$.)

Then $$k(f,g) \subseteq k(x,f,g) \subseteq k(x,y)$$ implies that $$[k(x,y):k(f,g)]$$ is a multiple of $$[k(x,y):k(x,f,g)]$$.

If, for example $$\gcd(d_2,e_2)=1$$, then $$[k(x,y): k(x,f,g)]=1$$, so there is no information on $$[k(x,y):k(f,g)]$$: " $$[k(x,y):k(f,g)]$$ is a multiple of $$1$$ ".

Also, $$k(f,g) \subseteq k(x,f,g) \subseteq k(x,y)$$ implies that $$[k(x,y):k(f,g)]$$ is a multiple of $$[k(x,f,g):k(f,g)] \leq \deg m(x, k(f,g))$$, the degree of the minimal polynomial of $$x$$ over $$k(f,g)$$.

Any comments are welcome! Thank you.

• A non-counterexample: $f=xy^2,g=xy^4$. $[k(x,y):k(f,g)=[k(x,y):k(x,y^2)=2$ Commented Aug 1 at 16:13