Sturm's method in counting roots of random polynomial equations (Q596503)

Let \(F_{n}(x,\omega )=a_{0}(\omega )+a_{1}(\omega )x+\dots+a_{n}(\omega )x^{n}\) be a polynomial, whose coefficients are real-valued random variables with given joint density function. Let \(N_{n}(B,\omega )\) be the number of zeros of the polynomial \(F_{n}(\cdot ,\omega )\) that belong to an interval \(B=(l,d)\). This paper is devoted to the problem of finding the probability distribution of the random variable \(N_{n}(B,\cdot )\). The authors describe the algorithm, based on the Sturm method, which enables us to express the probabilities \(P\{ N_{n}(B,\omega )=k\}\), \(k=0,1,\dots,n\), as multiple integrals. For special cases of random quadratic polynomials and polynomials of the form \(F^{*}(x,\omega )=a(\omega )x^{n}+b(\omega )x+c(\omega )\), they derive explicit formulas. The paper is concluded with some numerical results.