Small zeros of quadratic forms

Abstract:

We give upper and lower bounds for zeros of quadratic forms in the rational, real and $p$-adic fields. For example, given $r > 0$, $s > 0$, there are infinitely many forms $\mathfrak {F}$ with integer coefficients in $r + s$ variables of the type $(r,s)$ (i.e., equivalent over ${\mathbf {R}}$ to $X_1^2 + \cdots + X_r^2 - X_{r + 1}^2 - \cdots - X_{r + s}^2$ such that every nontrivial integer zero ${\mathbf {x}}$ has $|{\mathbf {x}}| \gg {F^{r/2s}}$, where $F$ is the maximum modulus of the coefficients of $\mathfrak {F}$.