Algebraic lattices and Pisot polynomials (Lenny Fukshansky, CMC)

A Z-module M in a number field K gives rise to a lattice in the corresponding Euclidean space via Minkowski embedding. Such lattices often carry inherited structure from the number field in question and can be attractive from both, theoretical and applied perspectives. We consider this construction when M is spanned by the set of roots of an irreducible polynomial f(x) of prime degree n. In this case, the resulting lattice has rank n or n-1 and includes the Galois group of f(x) as a subgroup of its automorphism group. Of particular interest is the case of Pisot polynomials, i.e., polynomials with one positive real root and the rest of the roots in the unit circle. We construct infinite families of such polynomials of any prime degree for which the resulting lattices have bases of minimal vectors, a property of interest in coding theory and cryptography applications. In case of the Galois group being cyclic, A_n, or S_n we derive formulas for the determinant of the lattice in terms of the symmetric functions of the roots of f(x). This is joint work with Evelyne Knight (Pomona College).