Bad pairs of polynomial zeros

Abstract

If an arithmetic progression F of length 2n and the number k with 2k≤n are given, can we find two monic polynomials with the same degrees whose set of all zeros form F such that both the number of bad pairs and the number of nonreal zeros are 2k? We will consider the case that both the number of bad pairs and the number of nonreal zeros are two. Moreover, we will see the fundamental relation between the number of bad pairs and the number of nonreal zeros, and we will show that the polynomial in x where the coefficient of xk is the number of sequences having 2k bad pairs has all zeros real and negative.