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 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 by $B$ iff $\dots$?