Small Values of Indefinite Diagonal Quadratic Forms at Integer Points in at least five Variables

TL;DR

This paper provides effective bounds for small non-zero solutions to indefinite diagonal quadratic forms in at least five variables, extending previous results and contributing to the quantitative Oppenheim conjecture.

Contribution

It extends Schlickewei's bounds on small zeros of quadratic forms to diagonal forms in five or more variables, with effective estimates and optimal dependence on form signature.

Findings

Derived effective estimates for small solutions to quadratic inequalities

Extended Birch and Davenport's approach to higher dimensions

Achieved bounds depending on the form's signature with negligible growth

Abstract

For any $\varepsilon > 0$ we derive effective estimates for the size of a non-zero integral point $m \in \mathbb{Z}^d \setminus \{0\}$ solving the Diophantine inequality $\lvert Q[m] \rvert < \varepsilon$, where $Q[m] = q_1 m_1^2 + \ldots + q_d m_d^2$ denotes a non-singular indefinite diagonal quadratic form in $d \geq 5$ variables. In order to prove our quantitative variant of the Oppenheim conjecture, we extend an approach developed by Birch and Davenport [BD58b] to higher dimensions combined with a theorem of Schlickewei [Sch85]. The result obtained is an optimal extension of Schlickewei's result, giving bounds on small zeros of integral quadratic forms depending on the signature $(r,s)$, to diagonal forms up to a negligible growth factor.