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Let $f,g \in \mathbb{C}[x,y]$ with total degrees $\deg_{1,1}(f),\deg_{1,1}(g) \geq 1$.

Write, $f=a_ny^n+a_{n-1}y^{n-1}+\cdots+a_1y^1+a_0$ and $g=b_my^m+b_{m-1}y^{m-1}+\cdots+b_1y^1+b_0$, for some $n,m \in \mathbb{N}$, $a_i, b_j \in \mathbb{C}[x]$.

Assume that the following four conditions $C_1,C_2,C_3,C_4$ are satisfied:

  • $C_1$: $\mathbb{C}(x,f,g)=\mathbb{C}(x,y)$, in other words, $\{x,f,g\}$ generate all $\mathbb{C}(x,y)$.
  • $C_2$: $\mathbb{C}(f,g) \subsetneq \mathbb{C}(x,y)$.
  • $C_3$: $\mathbb{C}(x,f) \subsetneq \mathbb{C}(x,y)$.
  • $C_4$: $\mathbb{C}(x,g) \subsetneq \mathbb{C}(x,y)$.

For example: $f=x+y^3, g=x+xy^2$ satisfy all 4 conditions. More generally, $f=f_1(x)+f_2(x)y^{3+L},g=g_1(x)+g_2(x)y^{2+L}$, $L \in \mathbb{N}$, $f_1,f_2,g_1,g_2 \in \mathbb{C}[x]$, each of degree $\geq 1$ is a family of $\{f,g\}$ satisfying the 4 conditions.

Questions:

(i) Is it possible to find 'all' possible forms of such $f$ and $g$? I guess that $f=f_1(x)+f_2(x)y^{3+L},g=g_1(x)+g_2(x)y^{2+L}$ was just one plausible form?

(ii) A special case, where we further assume that $y | g$, namely $b_0=0$. Perhaps it is easier to find an answer in this special case?

Non-examples:

(1) $f=(x+x^2)y, g=xf$: $C_2,C_3,C_4$ do not hold.

(2) $f=x^2y, g=x^3$: $C_2,C_3$ do not hold.

(3) $f=h(x), g=y$, where $h(x) \in \mathbb{C}[x]$ with $\deg_x(h) \geq 2$: $C_4$ does not hold.

Asked here, slightly more elaborately, without receiving comments yet.

Thank you very much!

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  • $\begingroup$ We can denote $t:=y^2$, then $s_1,s_2 \in \mathbb{C}[x,y^2]=\mathbb{C}[x,t]$. Then $\mathbb{C}(x,s_1,s_2)=\mathbb{C}(x,y^2)=\mathbb{C}(x,t)$. $\endgroup$
    – user237522
    Commented Jul 28 at 17:36
  • $\begingroup$ $y^2 \in \mathbb{C}(x,y^2)=\mathbb{C}(x,s_1,s_2)$, so $y^2 \in \mathbb{C}(s_1(0,y),s_2(0,y))$. $\endgroup$
    – user237522
    Commented Jul 28 at 17:41
  • $\begingroup$ I don't understand the relevance of $y$ and $\beta$. Why not immediately express everything in terms of $x$ and $t=y^2$? $\endgroup$
    – Dave Benson
    Commented Jul 29 at 11:02
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    $\begingroup$ @LSpice, thank you for your comment. I agree with you that I have a lot of similar questions to the one above and should tell about the motivation behind them. I will explain in a day or two. Thank you. $\endgroup$
    – user237522
    Commented Aug 3 at 23:05
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    $\begingroup$ @LSpice, thank you for your good advice. I will try to apply it from now on. Thank you. $\endgroup$
    – user237522
    Commented Aug 4 at 11:28

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