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.
Categories: math.NT