I wrote: << Let c = (c_0,...,c_(n-1)) have the standard normal distribution on R^n, i.e., the c_k are independent real random variables, each with probability density given by d(x) = (1/(sqrt(2 pi)) exp(-x^2 / 2)) QUESTION: Find the probability f(n) that the polynomial P(x) = x^n + c_(n-1) x^(n-1) + ... + c_1 x + c_0 has all its roots real ?
But on second thought the measure this puts on the space of nth degree real polys does not seem the most natural. Here's a nicer one: Consider all real polynomials c_n x^n + ... + c_0, and identify any two of them if one is a nonzero constant factor times the other (i.e., if they have the same roots). Then the equivalence classes form P^n, n-dimensional projective space, which has a natural measure on it. Its double cover is the usual n-sphere S^n with its usual measure, with each polynomial corresponding to a pair of antipodal points. The polynomials of degree < n form a subset of measure 0. With this measure on nth degree polynomials, it looks feasible to get the probability of having all real roots at least for some low degrees. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele