What is the probability of 3 real roots for a cubic equation?

Question

Several weeks ago a problem appeared here regarding the probability that a specified cubic equation would have 3 real roots when the coefficients were regarded as random variables. This is a follow-up and generalization of that problem. A general cubic equation y=Ax³+Bx²+Cx+D can be transformed into one with leading coefficient of one by division by A and then into the form x³+px+q by reducing the roots by B/3A. Consequently only this form needs to be considered and the question becomes: if the joint density function of p and q [call it f(p,q)] is known, what is the probability that the cubic equation has 3 real roots?

Answer 1

In Cardan's solution of the general cubic the expression D=81q²+12p³ behaves as a discriminant. If D>0, then there is one real root and a pair of complex conjugate roots. If D=0, there are 3 real roots, one of which is of multiplicity 2. If D<0, then there are 3 real roots and the equation is called an irreducible cubic because the roots cannot be found in closed form by algebraic means.

If the implicit expression D=0 is plotted (with q in the y direction and p in the x direction), the graph will have the shape of an right-pointing arrow with its tip at the origin and the graph will be symmetric about the x-axis. The probability of 3 real roots will be the probability that p and q fall inside the arrowhead.

For example if p and q are independent, i.e. f(p,q)=f₁(p)*f₂(q) and

f₁(p)=1/(b-a) for p in the interval [a,b] and 0 elsewhere and

f₂(q)=1/(d-c) for q in the interval [c,d] and 0 elsewhere, then the required probability will be the fraction of the area of the rectangle with diagonal endpoints (a,c) and (b,d) which lies within the arrowhead.