On the van der Waerden conjecture and zeros of polynomials

Abstract

Let F(x) = ∑ k=o n n k A kx k A n ≠ 0, and G(x) = ∑ k=o n n k B kx k B n ≠ 0, be polynomials with real zeros satisfying A n −1 = B n −1 = 0, and let H(x) = ∑ k=o n-2 n k A kB kx k. Using the recently proved validity of the van der Waerden conjecture on permanents, some results on the real zeros of H( x) are obtained. These results are related to classical results on composite polynomials.