Artin's conjecture is that for $f$ a separable polynomial irreducible modulo $p$ of degree 3, the equation $y^2 \equiv f(x) \mod p$ has a number of solutions $n$ satisfying $$n = p + O(\sqrt{p}).$$
I would like to understand better this $p$: is there a way to see that there are generically these $p$ expected solutions?
There are $k$ distinct values of $x$ where $f(x)=0$ which contributes $k$ points when $y=0$. On the other hand there are $p-k$ values of $x$ where $f(x) \ne 0$. Now we can roughly estimate that half of the time $f(x)$ is a perfect square for these $p-k$ values. This gives us $\frac{p-k}{2}$ solutions. Except since $y$ and $-y$ both contribute a solution we really have twice that many solutions. In total that means we have,
$$k+2*\frac{p-k}{2} = p$$
solutions naively speaking.