Communications of the
Korean Mathematical Society CKMS

If an arithmetic progression $F$ of length $2n$ and the number $k$ with $2k \leq n$ 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 $x^k$ is the number of sequences having $2k$ bad pairs has all zeros real and negative.

Keywords: bad pairs, good pairs, zeros, polynomials

MSC numbers: Primary 30C15; Secondary 11B25