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Hasse--Minkowski theorem 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 $\dots$?

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 $\dots$?

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 $\dots$?

Hasse--Minkowski theorem 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 $\dots$?

Hasse--Minkowski theorem 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 $\dots$?

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 $\dots$?