Do there exist polynomials $f,g$ such that $\mathbb{C}[a,b,c]\le\mathbb{C}[f,g]$ for $a,b,c$ given polynomials?

I want to prove something bigger than the problem in the title and I want to create a lemma that is useful for the solution of the problem. But I am unable to prove (or give a counterexample) the "lemma":

Suppose that $a,b,c$ are given polynomials in $\mathbb{C}[x]$ such that $\mathbb{C}[a,b,c] \subsetneq \mathbb{C}[x]$. There exist polynomials $f,g \in \mathbb{C}[x]$ such that $\mathbb{C}[a,b,c]\le\mathbb{C}[f,g]$ where $\mathbb{C}[f,g]\subsetneq \mathbb{C}[x]$? (The symbol $\le$ obviously means subring.)

Does anyone have any idea to prove this if it is true? Or give a counterexample if it is false?

Solution 1:

Consider the set of proper sub algebras $\mathcal{O}$ of $\mathbf{C}[x]$ which contain $a$, $b$, and $c$. Since the codimension of $\mathbf{C}[a, b, c]$ in $\mathbf{C}[x]$ is finite, we can choose $\mathcal{O}$ a proper sub algebra maximal with respect to inclusion.

Claim: any maximal proper sub algebra $\mathcal{O}$ is isomorphic as a $\mathbf{C}$-algebra to either $\{f \in \mathbf{C}[x] \mid f(0) = f(1)\}$ or to $\mathbf{C}[x^2, x^3]$.

You can probably prove this claim with elementary methods. (Hint: show that the codimension of $\mathcal{O}$ in $\mathbf{C}[x]$ is $1$ and then analyze what this means).

However, I would prove this by looking at the singularities of $\text{Spec}(\mathcal{O})$. If there are two singular points, then we can normalize one of them and we see that $\mathcal{O}$ is not maximal. If the $\delta$-invariant of the singularity is $> 1$, then we can add a local function without smoothing $\text{Spec}(\mathcal{O})$. Finally, if $C$ is an affine algebraic curve with a unique singular point of $\delta$-invariant $1$ and normalization equal to the affine line $\mathbf{A}^1 = \text{Spec}(\mathbf{C}[x])$, then it is isomorphic to the spectrum of one of the two rings in the claim.

This answers the question positively as the two example algebras can be generated by two elements as $\mathbf{C}$-algebras.