If $P_n=\{y\in M(n,\Bbb R)|$ $y$ is positive definite and symmetric $\}$ then for a fixed $y \in P_n$ consider the set $A=\{y[u]|u \in \Bbb Z^n\}$ where $y[u]=u^tyu$ is clearly positive. Now, how to prove that $\exists u_1\in P_n$ s.t $y[u_1]$ is minimal element of the set $A$ and there are finitely many $u_1=(g_1,...,g_n) \in \Bbb Z^n$ s.t $\gcd(g_1,...,g_n)=1$
This has a relation to my last question but being lengthier, it attracts fewer people which drives me to break it down. Maybe it helps!!
everything is in the excerpts I put at the last question. Given a matrix $y,$ square and real and symmetric and positive definite, there is a minimal eigenvalue $\varepsilon > 0.$ Given a column vector of integers $u,$ we know that $u^T y u \geq \varepsilon u \cdot u,$ using the usual dot product. Therefore, given some upper bound $M > 0,$ we find that the set of $u$ with $u^T y u \leq M $ is a subset of the set of $u$ with $u \cdot u \leq \frac{M}{\varepsilon} \; ,$ and is therefore finite as all entries of $u$ are integers. As a result, we check a finite set of integer vectors, and find that the minimum of the form really is achieved (despite lack of general compactness). Furthermore, setting the minimum itself as a non-strict upper bound, we find that it is achieved at a finite number of integer vectors.
This is very different from indefinite (real) forms. With $(x,y)$ integer pairs, $|x^2 - \pi y^2 |$ does not achieve its infimum; take $\frac{x}{y}$ as a continued fraction convergent for $\sqrt \pi$
Read the excerpts.