One-dimensional case: Let $s,k \in \mathbb{C}[x]$ with $s$ an even polynomial (degrees of all monomials are even) and $k$ an odd polynomial (degrees of all monomials are odd).
Denote $R=\mathbb{C}[s,k]$.
Assume that the field of fractions of $R$ is $\mathbb{C}(x)$.
Question: Is it possible to characterize all $s,k$ such that $\mathbb{C}[s,k] \subseteq \mathbb{C}[x]$ is flat?
For example: $s=x^2, k=x^3$; it is well-known that $\mathbb{C}[x^2,x^3] \subseteq \mathbb{C}[x]$ is not flat.
Answer: Notice that $R \subseteq \mathbb{C}[x]$ is an integral extension. Therefore, if $R \subseteq \mathbb{C}[x]$ is flat, then by a known result it is faithfully flat (see this). Then, since we assumed that the field of fractions of $R$ is $\mathbb{C}(x)$, we obtain that $R= \mathbb{C}[x]$ (see this).
Therefore, flatness of $R \subseteq \mathbb{C}[x]$ implies that $k=x$. Conversly, if $k=x$ then $R=\mathbb{C}[x] \subseteq \mathbb{C}[x]$ is flat. Concluding that: $\mathbb{C}[s,k] \subseteq \mathbb{C}[x]$ is flat if and only if $k=x$.
Am I right or missing something?
Two-dimensional case: Let $\beta: x \mapsto x, y \mapsto -y$; of course, $\beta$ is an involution on $\mathbb{C}[x,y]$ (= an automorphism of order two). Let $s_1,s_2,k_1,k_2 \in \mathbb{C}[x,y]$ with $s_1,s_2$ symmetric w.r.t. $\beta$ and $k_1,k_2$ skew-symmetric w.r.t. $\beta$.
Denote $R=\mathbb{C}[s_1,s_2,k_1,k_2]$.
Assume that the field of fractions of $R$ is $\mathbb{C}(x,y)$, namely, $\mathbb{C}(s_1,s_2,k_1,k_2)=\mathbb{C}(x,y)$.
Question: Is it possible to characterize all $s_1,s_2,k_1,k_2$ such that $\mathbb{C}[s_1,s_2,k_1,k_2] \subseteq \mathbb{C}[x,y]$ is flat?
Notice that $R \subseteq \mathbb{C}[x,y]$ may not be an integral extension (it is an algebraic extension, though).
Perhaps it is better to assume that each two of $\{s_1,s_2,k_1,k_2\}$ are algebraically independent over $\mathbb{C}$, in order to exclude cases such as $\mathbb{C}[x^2,x^3,xy,y]$ for which the extension is not flat (similar argument as above; flatness+integrality would imply faithful flatness, etc.).
Perhaps considering $s_i, k_j$ as elements of $(\mathbb{C}[x])[y]$ or of $(\mathbb{C}(x))[y]$ would help, though one has to be careful (integrality is applied to $R \subseteq \mathbb{C}(x)[y]$).
Any comment is welcome! Thank you.
