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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.

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This is much too complex a question. Firstly, the discriminant. It is a theorem of Mahler that the discriminant of a polynomial $P$ of degree $d$ is bounded above by $$d^d L(P)^{2d-2},$$ where $L(P)$ is the $L^1$ norm of the coefficient vector, which gives a bound of $$d^d (d+1)^{2d-2} H^{2d-2},$$ in terms of naive height. This bound is close to sharp - you can read Mahler's paper (free online).

Mahler, K., An inequality for the discriminant of a polynomial, Mich. Math. J. 11, 257-262 (1964). ZBL0135.01702.

Secondly, it is an empirical fact (which should be provable, but I don't have a proof at the moment), that for any $H$ (including $H=1$) the distribution of discriminant is lognormal, with mean about $1/2$ of the Mahler bound (this is for height $1$), and very small variance (standard deviation is around $20$ for degree $200$ and around $30$ for degree $400,$ indicating sublinear growth (or, if you prefer, possibly linear growth of variance). This should give you a reasonable estimate of the number of polynomials with discriminant whose log is not too far from the mean. For very large discriminant, we are in the world of large deviations, and I have no idea what to guess.

For resultants, I don't know the analogue of Mahler's bound (I am guessing his method can be adapted). The distribution again seems log-normal, but with smaller mean (about half the mean of the discriminant) and larger standard deviation.

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  • $\begingroup$ Igor, I am well aware of this paper of Mahler. I think there is a typo in your upper bound: it should actually be $d^dL(P)^{2d-2}$ and consequently $d^d(d+1)^{2d-2}H^{2d-2}$. $\endgroup$ – Anton Jun 2 '17 at 16:46
  • $\begingroup$ @Anton yes, you are right re typo. $\endgroup$ – Igor Rivin Jun 2 '17 at 16:50
  • $\begingroup$ @Anton but I used it in my comment on the mean of the log, which, in reality, is almost exactly half of the log of the Mahler bound. $\endgroup$ – Igor Rivin Jun 2 '17 at 17:05
  • $\begingroup$ Thanks Igor! After playing around with data sets on some special cases I can see that it is lognormal. I am pretty sure that the resultant case is very similar. It would be very interesting to see the proof of this fact though. $\endgroup$ – Anton Jun 2 '17 at 23:19

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