Random polynomials with a prescribed number of real zeros

摘要

Consider a polynomial of the form a_0 + a_1x + …+a+nx~n with random coefficients a_j. It is shown that, under mild conditions, one can choose the distributions of the a_j from a given class of distributions so that, with probability arbitrarily close to 1, the random polynomial has a prescribed number of real roots. The proof is based on the gliding hump method. It is also shown how this method can be used to solve related problems for random sums of orthogonal polynomials and random power series.