What is the likelihood that a degree 3 monic polynomial will have 3 real roots.

Question

The discriminant of a monic cubic polynomial $x^3 + bx^2 + cx + d $ is given by the expression $b^2 c^2 - 4 c^3 - 4 b^3d + 18 bcd - 27 d^2$. If this expression is positive then all the roots are real. Right? I did some experiments in Mathematica with random monic cubics whose coefficients are in the interval $[-n,n]$. It appears that as $n$ gets bigger the likelihood that the discriminant is positive increases. Here is a list of the number of random monic cubics with positive discriminant from a sample of 1000 where the coefficients are in the interval $[-10^k, 10^k]$ for $k = 1,2,...,15$. $\{432, 560, 601, 620, 616, 634, 638, 627, 620, 618, 631, 617, 618, 630, 648\}$. Does the likelihood go to some limit?

Answer 1

The dominant terms are $b^2c^2$ and $4b^3d$ because they are the only terms that are around $N^4$, instead of $N^3$ or smaller.
The chance that $b^2c^2-4b^3d>0$ equals the chance that $c^2>4bd$.
This includes the $50\%$ chance that $bd<0$.
Now suppose $b,c,d$ are all positive. Let $b=Nx$, $c=Nz$, $d=Ny$, so $x,y$ and $z$ are all between $0$ and $1$.
First, suppose $b$ and $d$ are fixed. The chance that $c^2>4bd$ equals the chance that $2\sqrt{bd}<c<N$, or $2\sqrt{xy}<z<1$. It equals zero if $2\sqrt{xy}>1$, otherwise it equals $1-2\sqrt{xy}$.
To get the overall probability, I average over all possible values of $b$ and $d$. Equivalently, I integrate over $x$ and $y$.
$$B=\int_0^1\int_0^1 \max(0,1-2\sqrt{xy})\, dx\,dy\\ =\int_0^{1/4}\int_0^11-2\sqrt{xy}\,dx\,dy+\int_{1/4}^1\int_0^{\frac1{4y}}1-2\sqrt{xy}\,dx\,dy\\ =\int_0^{1/4}1-\frac43\sqrt{y}\,dy+\int_{1/4}^1\frac1{4y}-\frac43\sqrt{\frac1{64y^2}}\,dy\\ =\frac14-\frac19+\frac1{12}\log(y)_{1/4}^1=\frac5{36}+\frac{\log2}6$$ I broke it into two pieces at $y=1/4$ because $1-2\sqrt{xy}$ is always positive if $y<1/4$ and $0<x<1$.
The overall chance is then $50\%+50\%B=0.6272$.
Remark: This is the same value as Travis' calculation The probability that a random (real) cubic has three real roots for a general quadratic where the leading coefficient is also between $-N$ and $N$. I expect that is because the $x^3$ term tends to be negligible if $x$ is one of the two roots of the quadratic.