Maria Fernanda Zordan Bonini

arXiv2026

On The Cyclicity of Algebraic Lattices

This work presents theoretical advances in the study of cyclic and quasi-cyclic lattices. First, we provide elementary facts regarding cyclic and quasi-cyclic lattices, and discuss about the cyclicity of some notable lattices. The main contributions of the paper is in the algebraic setting: we investigate cyclic lattices arising from $\mathbb{Z}$-modules in Galois number fields via the Minkowski embedding. We establish necessary and sufficient conditions for an algebraic lattice to be cyclic over both cyclic and general Galois number fields, expressed in terms of naturally associated groups. Moreover, we derive a necessary and sufficient condition for ideal lattices to be cyclic, depending on the factorization of the ideal in the underlying number field.

math.NT↗

arXiv2026

On permutation-invariant construction of glued lattices

Given a permutation $τ$ on $n$ letters, we consider lattices spanned by an orbit of one vector $\boldsymbol x$ in $\mathbb R^n$ under the action of $τ$ by permutation of the coordinates. Such lattices generalize the important class of cyclic lattices and have previously been studied in~\cite{perm}, where a bound on their rank was established. We prove a sufficient condition on $\boldsymbol x$ for this bound to be achieved. We further investigate the structure of such permutation-invariant lattices, proving that they are glued by the permuted vector from the orthogonal cyclic blocks and giving a determinant formula for the lattice in terms of determinants of these blocks and the norm of the permuted vector. In the case $\boldsymbol x$ is an integer vector, these blocks are sublattices of the root lattices $A_k$ in respective dimensions with root lattices themselves and their glued direct sums also realizable by this construction. We also exhibit a glued construction of permutation-invariant algebraic integral lattices from collections of cyclic number fields. Finally, we prove a strengthened version of a previous result of~\cite{lf_ek} on a related construction of well-rounded lattices spanned by sets of algebraic conjugates.

math.NT↗