Height bounds on zeros of quadratic forms over $\overline{\mathbb Q}$

TL;DR

This paper establishes explicit height bounds for small zeros of quadratic forms and systems over algebraic numbers, extending classical results and providing new bounds for various quadratic and linear polynomial systems.

Contribution

It provides new explicit height bounds for small zeros of quadratic forms and systems over algebraic numbers, generalizing previous classical results.

Findings

Bound on height of smallest zero outside an algebraic set for a single quadratic form.

Existence of small-height simultaneous zeros for systems of quadratic forms.

Explicit height bounds for zeros of inhomogeneous quadratic and linear polynomial systems.

Abstract

In this paper we establish three results on small-height zeros of quadratic polynomials over $\overline{\mathbb Q}$. For a single quadratic form in $N \geq 2$ variables on a subspace of $\overline{\mathbb Q}^N$, we prove an upper bound on the height of a smallest nontrivial zero outside of an algebraic set under the assumption that such a zero exists. For a system of $k$ quadratic forms on an $L$-dimensional subspace of $\overline{\mathbb Q}^N$, $N \geq L \geq \frac{k(k+1)}{2}+1$, we prove existence of a nontrivial simultaneous small-height zero. For a system of one or two inhomogeneous quadratic and $m$ linear polynomials in $N \geq m+4$ variables, we obtain upper bounds on the height of a smallest simultaneous zero, if such a zero exists. Our investigation extends previous results on small zeros of quadratic forms, including Cassels' theorem and its various generalizations and contributes to the literature of so-called "absolute" Diophantine results with respect to height. All bounds on height are explicit.