Let $f=f(t) \in \mathbb{C}[t]$ be a non-scalar polynomial. Write $f=a_{2n}t^{2n}+a_{2n-1}t^{2n-1}+\cdots+a_2t^2+a_1t+a_0$, where $a_j \in \mathbb{C}$.
The map $\iota: t \mapsto -t$ is an involution on $\mathbb{C}[t]$, namely, a $\mathbb{C}$-algebra automorphism of degree two.
Denote the set of symmetric elements w.r.t. $\iota$ by $S$ and the set of skew-symmetric elements w.r.t. $\iota$ by $K$. Notice that even polynomials are symmetric elements w.r.t. $\iota$, while odd polynomials are skew-symmetric elements w.r.t. $\iota$.
Write $f=s+k$, where $s \in S$ and $k \in K$.
Clearly, $\mathbb{C}[f] \subseteq \mathbb{C}[t]$ is a UFD ($\mathbb{C}[f]$ is isomorphic to $\mathbb{C}[t]$).
Question 1: Is it possible to find a general form of $f$ such that $R_{f}:=\mathbb{C}[s,k]$ is a UFD?
Partial answer: In the following four cases $R_{f}$ is a UFD:
(i) $k=0$, so $f=s \in S$, and then $R_{f}=\mathbb{C}[s,0]=\mathbb{C}[s] \cong \mathbb{C}[t]$.
(ii) $s=0$, so $f=k \in K$, and then $R_{f}=\mathbb{C}[0,k]=\mathbb{C}[k] \cong \mathbb{C}[t]$.
(iii) $k=t$, so $R_{f}=\mathbb{C}[s,t]=\mathbb{C}[t]$.
(iv) $s=k^{2m}$ for some $m$, so $R_{f}=\mathbb{C}[k^{2m},k]=\mathbb{C}[k] \cong \mathbb{C}[t]$. For example, $s=\lambda t^6$, $k=\mu t^3$, so $R_{f}=\mathbb{C}[\lambda t^6,\mu t^3]=\mathbb{C}[t^3] \cong \mathbb{C}[t]$.
Question 1': Are there additional cases?
Non-example: For $f=t^3+t^2$, $R_{f}=\mathbb{C}[t^2,t^3]$ is not a UFD, as $t^2t^2t^2=t^3t^3$ shows.
Remark: This question is relevant, especially the comments of Mohan and user26857, saying the following: "If $R$ is Noetherian, integrally closed and dimension one $\mathbb{C}$-subalgebra of the polynomial ring, it is isomorphic to $\mathbb{C}[t]$". Therefore, here, if $R_{f}$ is not a UFD, then it is not isomorphic to $\mathbb{C}[t]$, hence $R_{f}$ is not integrally closed.
On the other hand, if $R_{f}$ is integrally closed, then it equals its integral closure, which is of the form $\mathbb{C}[g]$ for some $g \in \mathbb{C}[t]$ (= this result appears in the paper of Paul Eakin "A note on finite dimensional subrings of polynomial rings"). Therefore, $R_{f}$ is isomorphic to $\mathbb{C}[t]$, hence it is a UFD.
Summarizing, my questions 1 and 1' are equivalent to the following question:
Question 1'': Let $s \in S$ (= even polynomials), $k \in K$ (= odd polynomials). Is it possible to find general forms of $s$ and $k$ such that $R_{s+k}=\mathbb{C}[s,k]$ is integrally closed? (Are cases (i), (ii) and (iii) the only ones?).
Important remark to question 1'': This question and its answer are highly relevant.
Question 2: I guess that questions 1 and 1' are not too difficult (perhaps some algebraic geometry is needed?), so what if we replace $\mathbb{C}$ by an arbitrary UFD $D$?
Any comments are welcome! Thank you.
