Integral orthogonal bases of small height for real polynomial spaces

Let \( \mathcal{P}_N(\mathbb{R}) \) be the space of all real polynomials in \( N \) variables with the usual inner product \( \langle \cdot, \cdot \rangle \) on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form representing this inner product on the space of coefficient vectors of all polynomials in \( \mathcal{P}_N(\mathbb{R}) \) of degree \( \leq M \). We exhibit two applications of this formula. First, given a finite-dimensional subspace \( V \) of \( \mathcal{P}_N(\mathbb{R}) \) defined over \( \mathbb{Q} \), we prove the existence of an orthogonal basis for \( (V, \langle \cdot, \cdot \rangle) \), consisting of polynomials of small height with integer coefficients, providing an explicit bound on the height; this can be viewed as a version of Siegel’s lemma for real polynomial inner product spaces. Secondly, we derive a criterion for a finite set of points on the unit sphere in \( \mathbb{R}^N \) to be a spherical \( M \)-design.