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Well-rounded lattices in R² via the canonical homomorphism and the twisted homomorphism

Grant number: 23/15735-8
Support Opportunities:Scholarships in Brazil - Scientific Initiation
Effective date (Start): March 01, 2024
Effective date (End): February 28, 2025
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Algebra
Principal Investigator:Carina Alves
Grantee:Nicoll Vanessa Jerez Nieves
Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil

Abstract

One of the techniques for generating lattices and evaluating their packing density is through the application of certain homomorphisms to certain free Z-modules of rank n contained in a number field K of degree n. The lattices generated by this method are known as algebraic lattices.The advantage of obtaining lattices using this method is that we can identify the lattice points in Rn with the elements of K. In this way, we can use some properties of the field K, which have a richer algebraic structure, in the study of such lattices.Well-rounded lattices are those in which the set of vectors with minimum norm generate the entire space. They arise in different contexts, including packaging problems, contact number problems, discrete optimization problems, applications in code theory, Minkowski conjecture, among others.In this context, the following question arises: which algebraic lattices are well rounded? Motivated by this question and the applicability of well-rounded lattices, we studied the construction of algebraic lattices via quadratic fields and analyzed which lattices are well-rounded via the canonical and twisted homomorphism.

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