I've been trying to work through Coutinho's Primer on Algebraic D-Modules and I'm having trouble with proving the following exercise:
Let $P \in A_1 (\mathbb{C})$, the first Weyl Algebra over $\mathbb{C}$, $f \in \mathbb{C} [x]$
Show that the vector space of polynomial solutions to $P(f) = 0$ has finite dimension over $\mathbb{C}$
A solution would be really appreciated, thanks.
A subspace of $\mathbb{C}[x]$ is finite-dimensional iff the degrees of elements in it are bounded, so let's try to prove a bound on the degrees. Write $P = \sum a_{i, j} x^i D^j$ where $D = \frac{\partial}{\partial x}$. Then
$$P(x^n) = \sum a_{i, j} n (n - 1) ... (n - j + 1) x^{n + i - j}.$$
The leading term of this polynomial in $x$ is controlled by the largest value of $i - j$ such that $a_{i, j} \neq 0$; this leading term in turn has coefficient a polynomial in $n$ whose leading term is controlled by the largest value $j_{\text{max}}$ of $j$ which occurs in the previous leading term, and this is unique; call the corresponding $i$ for which $i - j$ has its largest value $i_{\text{max}}$. In particular, this coefficient is not identically zero as a polynomial in $n$ and hence will not vanish for sufficiently large $n$.
Now it follows that if $f$ has degree $n$ for $n$ sufficiently large then $P(f)$ has degree $n + i_{\text{max}} - j_{\text{max}}$ and in particular cannot vanish.