Minkowski’s Fundamental Theorem and Its Applications

Given a real valued function \(F(x_{1},x_{2},\ldots ,x_{n})\) of real variables \(x_{1},x_{2},\) \(\ldots ,x_{n}\) , one of the basic problems in Number Theory is to find conditions under which there exist integers \(u_{1},u_{2},\ldots ,u_{n}\) not all zero such that \(|F(u_{1},u_{2},\ldots ,u_{n})|\le \lambda , \) where \(\lambda \) depends on certain invariants of F. Hermann Minkowski gave a systematic treatment of such problems by interpreting these arithmetical problems in geometrical language. He proved a general result known as Minkowski’s Fundamental Theorem and gave several important applications of it. A number of proofs of this theorem are known. In this chapter we give two proofs of Minkowski’s Fundamental Theorem and give some of its applications, in particular to positive definite quadratic forms and introduce Hermite’s constant \(\gamma _n\) . We also prove Minkowski’s Theorem on system of linear forms and give its application to simultaneous approximation of reals by rationals. We briefly define lattices and then prove Minkowski’s Fundamental Theorem for lattices. Generalizations of Minkowski’s Fundamental Theorem by Blichfeldt and Van der Corput are also given.