A Family of Polynomials Uniting the Plastic Constant, the Golden Ratio, and Salem Numbers

AbstractThis work investigates a family of polynomials constructed from a simple alternatingrecurrence with doubling coefficients. It is observed that the first polynomialcontains a root equal to the negative of the plastic constant. Symmetrization ofthe second polynomial yields two Salem numbers, the cyclotomic polynomial Φ10(related to the golden ratio), and a degree-14 polynomial with a rich root structure.For the third and fourth polynomials, the symmetrization produces families of highdegreepolynomials, most of whose roots lie exactly on the unit circle, even thoughthe polynomials themselves are not cyclotomic. All results are experimental andwere obtained with the computer algebra system Sage.