Let $f(x), g(x) \in \mathbb Z[x]$ be polynomials of positive degrees $r$ and $s$ respectively, and let $\operatorname{Res}(f,g)$ denote their resultant. Further, let $H(f)$ denote the naive height of a polynomial $f$; that is, $H(f)$ is equal to the maximum of absolute values of its coefficients.

Now let us fix positive integers $r, s, H_0, R_0$. Then there are $(2H_0)^2(2H_0+1)^{r}(2H_0+1)^{s}$ pairs of integer polynomials $(f,g)$ such that $\deg f = r$, $\deg g = s$, and $\max\{H(f),H(g)\}\leq H_0$. My question is: what proportion of those pairs of polynomials satisfy the inequality

$$|\operatorname{Res}(f,g)|\geq R_0?$$

I would like to derive a non-trivial lower bound on this quantity in terms of $r, s, H_0, R_0$. Was this question studied before? I would be thankful for any references.

Also, if we set $g(x)$ to be equal to the derivative of $f(x)$, then this question essentially reduces to the question about the magnitude of the absolute value of a discriminant $D(f)$ of a polynomial $f(x)$. Then the question can be rephrased as follows: what proportion of polynomials $f(x)$ of degree $r$ and height at most $H_0$ satisfy the inequality $|D(f)| \geq R_0$? Perhaps, there are references to this special case?

P.S. Note that it is a consequence of Hadamard's inequality that

$$|\operatorname{Res}(f,g)| \leq (r+1)^{s/2}(s+1)^{r/2}H(f)^sH(g)^r,$$

so we need to demand $R_0 \leq (r+1)^{s/2}(s+1)^{r/2}H_0^{s+r}$, for otherwise the answer would trivially be zero.