turkmath.org

Well-rounded lattices are vital in extremal lattice theory, since the classical optimization problems can usually be reduced to them. On the other hand, many of the important constructions of Euclidean lattices with good properties come from different algebraic settings. This prompts a natural question: which of the lattices coming from algebraic constructions are well-rounded? We consider three such constructions: ideal lattices from number fields, lattices from finite abelian groups and function fields, and cyclic lattices from quotient polynomial rings. In each of these cases, we provide a partial answer to the above question, as well as discuss some generalizations, applications, and directions for future research.