Minima of Positive Definite Quadratic Forms

In Chapter 2 we defined Hermite’s constant \(\gamma _n\) , the minima of positive definite quadratic forms, and obtained an upper bound on \(\gamma _n\) as an application of Minkowski’s Fundamental Theorem. In this chapter, we give a proof of Hermite’s inequality \(\gamma _n \le (\frac{4}{3})^{\frac{(n-1)}{2}}\) . We also give a proof of Mordell’s result \(\gamma _n \le \gamma _{n-1}^{(n-1)/(n-2)}\) . Further, we obtain \(\gamma _n\) for \(n=2\) and 3. As an application, we derive necessary and sufficient conditions for a positive integer to be a sum of k squares for \(k=2,3\) .