Odd degree polynomials on real Banach spaces

A classical result of Birch claims that for given k, n integers, n-odd there exists some N = N(k, n) such that for an arbitrary n-homogeneous polynomial P on IRN there exists a linear subspace Y hooked right arrow IRN of dimension at least k, where the restriction of P is identically zero (we say that Y is a null space for P). Given n > 1 odd, and arbitrary real separable Banach space X (or more generally a space with w*-separable dual X*), we construct an n-homogeneous polynomial P with the property that for every point 0 not equal x is an element of X there exists some k is an element of IN such that every null space containing x ha's dimension at most k. In particular, P has no infinite dimensional null space. For a given n odd and a cardinal tau, we obtain a cardinal N = N(T, n) = exp(n+1) tau such that every n-homogeneous polynomial on a real Banach space X of density N has a null space of density tau.