TL;DR
This paper reviews the development of the Geometry of Numbers, focusing on lattice properties and their applications in number theory, Diophantine equations, and functional analysis, highlighting Minkowski's foundational contributions.
Contribution
It provides a comprehensive overview of the major developments in the Geometry of Numbers and discusses the properties and applications of lattices in various mathematical fields.
Findings
Lattices are fundamental in studying algebraic numbers.
Geometry of Numbers connects geometric and algebraic properties.
Applications include solving Diophantine equations and functional analysis.
Abstract
In this paper we discuss about properties of lattices and its application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to study the properties of algebraic numbers. It has application on various other fields of mathematics especially the study of Diophantine equations, analysis of functional analysis etc. This paper will review all the major developments that have occurred in the field of geometry of numbers. In this paper we shall first give a broad overview of the concept of lattice and then discuss about the geometrical properties it has and its applications.
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