Number of rational functions with low-degree numerator and denominator

Question

The problem

Let $\mathbb F$ be a finite field and $d < | \mathbb F |$. (Can think of the setting $d \approx |\mathbb F|/2$)

Let $$ A = \left\{ \frac{p(x)}{q(x)} \ \colon \ p,q \in \mathbb F[x] \text{ and } \mathrm{deg}(q), \mathrm{deg}(p) \leq d \right\} \setminus \left\{p(x) \in \mathbb F \colon \mathrm{deg}(p) \leq d\right\}$$ That is, $A$ is the set of all univariate rational functions with both numerator and denominator polynomials of degree at most $d$, who are not low-degree themselves. Note that the degree of $q$ need not be smaller than that of $p$; for example, $1/x$ is an interesting function to us.

What is the cardinality of $|A|$?

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A definitional issue

Since I'm only interested in counting, I'll choose a special symbol $* \notin \mathbb F$ and let $\frac{p(x)}{q(x)} = *$ whenever $q(x) = 0$. If there is a more elegant or correct suggestion for dealing with $q(x)=0$ I'm happy to consider it.

Answer 1

Let $M_d$ be the set of all monic polynomials over $\mathbb{F}$ of degree (exactly) $d$, and let $$M_{\leqslant d}:=\bigcup_{k=0}^{d}M_d,\qquad G_d:=\{(p,q)\in M_{\leqslant d}^2 : \gcd(p,q)=1\}.$$ As any pair $(p,q)\in M_{\leqslant d}^2$ is uniquely represented by $(p'g,q'g)$, where $g=\gcd(p,q)$ is a monic polynomial, and $\gcd(p',q')=1$, we have $$|M_{\leqslant d}|^2=\sum\limits_{k=0}^{d}|M_k|\cdot|G_{d-k}|.$$ Denoting $K:=|\mathbb{F}|$, we compute $|M_d|=K^d$, $$|M_{\leqslant d}|=\frac{K^{d+1}-1}{K-1},\quad|G_d|=|M_{\leqslant d}|^2- K|M_{\leqslant d-1}|^2,\quad|G_0|=1.$$ Returning to your $A$, we can make $p$ and $q$ monic by carrying out leading coefficients, thus $$|A|=(K-1)(|G_d|-|M_{\leqslant d}|)=K^{d+1}(K^d-1).$$