I am trying to study a problem in algebraic number theory through a set of computational experiments. I have an enormous (say, of size $X$) family $\mathcal{F}$ of polynomials and I'm trying to understand how many distinct values a certain function $g$ takes over this family.
To experiment, I randomly sample $N$ (say $10^5$ at a time) polynomials from the $\mathcal{F}$ and compute $g(f)$ for each polynomial $f$ in my sample. I keep a list of each of these outputs $g(f)$ and count how many times each occurs. I can repeat this sampling and tallying process several times.
If I assume that $g(f)$ are uniformly distributed over the (finite) possible range of values, I should be able to guess the size of the image, $\# g(\mathcal{F})$ based on how many repetitions I see.
Is there a simple way to do this?
The inverse question is standard. That is, if I know that $\# g(\mathcal{F}) = Y$ and I take $N$ random draws with repetition from $Y$, then the expected number of repetitions can be obtained from a binomial distribution, as in this other question.
One approach for me would be to iteratively guess various $Y$ and see what looks close. But I feel like there must be a better way (and I have a lot to learn about fundamental statistical questions).
