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Integral Orthogonal Bases Of Small Height For Real Polynomial Spaces, Lenny Fukshansky
Integral Orthogonal Bases Of Small Height For Real Polynomial Spaces, Lenny Fukshansky
CMC Faculty Publications and Research
Let PN(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 PN(R) of degree ≤ M. We exhibit two applications of this formula. First, given a finite dimensional subspace V of PN(R) defined over Q, we prove the existence of an orthogonal basis for (V, < , >), consisting of polynomials of small height …