The Hasse-Minkowski theorem states that if

$$Q(x\_1,\ldots,x\_n) = \sum\_{i,j=1}^n a\_{ij} x\_ix\_j$$
is a quadratic form with $a\_{ij} \in \mathbb Z$ and $\det (a\_{ij}) \neq 0$, then the equation
$$Q(x\_1,\ldots,x\_n)=0$$
has a nontrivial solution in the integers if and only if it has a solution in the reals and all the $p$-adics.

> 
> 1. Can there be a bounded version to this statement such as if
>  $$Q(x\_1,\ldots,x\_n) = \sum\_{i,j=1}^n a\_{ij} x\_ix\_j$$
>  is a quadratic form with $a\_{ij} \in \mathbb Z$ and $\det (a\_{ij}) \neq 0$, then the equation
>  $$Q(x\_1,\ldots,x\_n)=0$$
>  has a nontrivial solution in the integers with each coordinate bound in absolute value by $B$ iff $\dots$?
> 2. Can there be a bounded version to at least a linear version of the statement such as if
>  $$L(x\_1,\ldots,x\_n) = \sum\_{i=1}^n a\_{i} x\_i$$
>  is a linear form with $a\_{i} \in \mathbb Z$ and **some criteria**, then the equation
>  $$L(x\_1,\ldots,x\_n)=0$$
>  has a nontrivial solution in the integers with each coordinate bound in absolute value by $B$ iff $\dots$?
> 
> 
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