#### Document Type

Article

#### Department

Mathematics (CMC)

#### Publication Date

2009

#### Abstract

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

*(R) of degree ≤*

_{N}*M*. We exhibit two applications of this formula. First, given a finite dimensional subspace

*V*of P

*(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*

^{N}*M*-design.

#### Rights Information

© 2009 Lenny Fukshansky

#### Terms of Use & License Information

#### Recommended Citation

Fukshansky, Lenny. "Integral Orthogonal Bases of Small Height for Real Polynomial Spaces." Online Journal of Analytic Combinatorics 4(2009). Web. 3 Apr. 2012.

## Comments

Please note that this article was first published by the University of Auckland in the

Online Journal of AnalyticCombinatorics, volume 4 (2009).