Bounded version of linear and quadratic Hasse--Minkowski theorem

Question

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 $\dots$?

Answer 1

The linear case is Siegel's lemma. I do not know the quadratic case: Cassels does not have it, perhaps O'Meara does. Note that the regulator is closely related where we seek a representation of $1$ by $x^2-dy^2$ in integers, and bounds on the regulator are hard.

Answer 2

For the first question, the answer is yes; you can find the statement and a proof in Cassels (pp 86-89). A far reaching generalization was obtained by J. Vaaler in 1987 concerning the height of a totally isotropic subspace of quadratic forms on subspace of $F^n$, where $F$ is a number field. Since then there have been various types of results like that. You may want to check out Lenny Fukshansky's papers. A recent paper of him with Chan and Henshaw generalizes Vaaler's result to zeros avoiding algebraic varieties.