Algebraic points of small height with additional arithmetic conditions

We treat a few related problems about the existence of algebraic points of small height that satisfy certain arithmetic conditions. All bounds on height of points in question are explicit. First we prove the existence of a small-height point over a fixed number field outside of a collection of subspaces; this includes a generalization and a converse of the celebrated Siegel’s Lemma, [5]. Next, assuming that a quadratic form has a zero outside of a collection of subspaces over a fixed number field, we prove the existence of such a zero of bounded height; this generalizes a result of Masser, [19]. A corollary of this is an extension of Cassels’ famous theorem on small zeros of quadratic forms (see [7]) to small non-singular zeros of quadratic forms. Finally, we prove a theorem about existence of small-height zeros of homogeneous polynomials of arbitrary degree over Q outside of a collection of subspaces. This direction is similar in spirit to the so-called “absolute” results like, for instance, the absolute version of Siegel’s Lemma of Roy and Thunder, [24].