A Simple Family of Characteristic Polynomials Yielding Negative Pisot Numbers

AbstractWe present an extremely simple infinite family of characteristic polynomials that producenegative Pisot numbers for every even degree. For n = 2k (k = 1, 2, 3, . . . ) thepolynomial isP2k(x) = x2k + x2k−1 − x2k−2 + x2k−3 − · · · + x − 1,i.e. the first two terms are positive, then the signs alternate, ending with −1. Numericalcomputations for degrees up to 40 show that each polynomial is irreducible, has a uniquenegative real root of modulus > 1, and all other roots lie inside the unit circle – hence itis a negative Pisot number. The moduli increase monotonically and tend to 2 from below.The evidence strongly suggests that the whole infinite family shares this property.