The exact number of real roots of the Bernoulli polynomials (Q420774)

This paper begins with a summary of earlier work by several authors who have investigated the number of real roots of the classical Bernoulli polynomials \(B_n(x)\).NEWLINENEWLINEThere are many cases for which the number of real roots cannot be determined exactly by these earlier methods. The present paper introduces a new method that relies on the fact that a Bernoulli polynomial is well approximated by a sinusoidal function on an interval about \(x = 1/2\), and that it `peels off' this sinusoidal in a predictable way. Apart from a minuscule fraction of cases, the new method calculates, in principle, the exact number of real roots of all Bernoulli polynomials for arbitrarily large index \(n\).