On lattices generated by algebraic conjugates of prime degree
Lenny Fukshansky, Evelyne Knight
TL;DR
The paper studies lattices $L_f$ obtained from the Minkowski embedding of the ${\mathbb Z}$-module generated by the conjugates $\alpha_i$ of a root $\alpha$ of a monic irreducible polynomial $f\in{\mathbb Z}[x]$, showing that the Galois group $G$ acts by lattice automorphisms and yields large symmetry groups. For prime degree $n$, the rank of $L_f$ is $n$ if $a_{n-1}\neq 0$ and $n-1$ if $a_{n-1}=0$, with well-roundedness guaranteed under certain bounds on the minimal norm and linear-form conditions; the authors provide explicit 2D and 3D WR criteria. They establish the existence of infinite families of well-rounded, nearly orthogonal lattices arising from large Pisot numbers, showing these lattices can contain a basis of minimal vectors and giving determinant formulas for several Galois types (cyclic, $S_n$, and $A_n$). The work connects algebraic number theory (Pisot polynomials, strong approximation) with lattice geometry, offering new, highly symmetric lattice families with potential applications in coding theory and cryptography and laying groundwork for extensions to composite degrees.
Abstract
We consider Euclidean lattices spanned by images of algebraic conjugates of an algebraic number under Minkowski embedding, investigating their rank, properties of their automorphism groups and sets of minimal vectors. We are especially interested in situations when the resulting lattice is well-rounded. We show that this happens for large Pisot numbers of prime degree, demonstrating infinite families of such lattices. We also fully classify well-rounded lattices from algebraic conjugates in the 2-dimensional case and present various examples in the 3-dimensional case. Finally, we derive a determinant formula for the resulting lattice in the case when the minimal polynomial of an algebraic number has its Galois group of a particular type.