Quadratic geometry of numbers

Abstract:

We give upper bounds for zeros of quadratic forms. For example we prove that for any nondegenerate quadratic form $\mathfrak {F}({x_1}, \ldots , {x_n})$ with rational integer coefficients which vanishes on a $d$-dimensional rational subspace $(d > 0)$ there exist sublattices ${\Gamma _0}, {\Gamma _1}, \ldots ,{\Gamma _{n - d}}$ of ${\mathbf {Z}^n}$ of rank $d$, on which $\mathfrak {F}$ vanishes, with the following properties: \[ {\text {rank}}({\Gamma _0} \cap {\Gamma _i}) = d - 1,\quad {\text {rank}}({\Gamma _0} \cup {\Gamma _1} \cup \cdots \cup {\Gamma _{n - d}}) = n\] and \[ {(\det {\Gamma _0})^{n - d}}\det {\Gamma _1} \cdots \det {\Gamma _{n - d}} \ll {F^{{{(n - d)}^2}}}\], where $F$ is the maximum modulus of the coefficients of $\mathfrak {F}$.