  Let $P_N(R)$ be the space of all real polynomials in $N$ variables with the
usual inner product $<, >$ 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 $P_N(R)$ of degree $\leq M$. We exhibit two applications of this
formula. First, given a finite dimensional subspace $V$ of $P_N(R)$ defined
over $Q$, we prove the existence of an orthogonal basis for $(V, <, >)$,
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 $R^N$ to be a spherical
$M$-design.
