The reciprocal algebraic integers having small houses

Abstract

Let $\alpha$ be an algebraic integer of degree $d$, which is reciprocal. The house of $\alpha$ is the largest modulus of its conjugates. We proved that $d$-th power of the house of reciprocal $\alpha$ has a limit point. We presented a property of antireciprocal hexanomials. We compute the minimum of the houses of all reciprocal algebraic integers of degree $d$ having the minimal polynomial which is a factor of a $D$-th degree reciprocal or antireciprocal polynomial with at most eight monomials, say $\mathrm{mr}(d)$, for $d$ at most 180, $D\le 1.5d$ and $D\le 210$. We show that it is not necessary to take into account unprimitive polynomials. The computations suggest several conjectures.