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 [InlineMediaObject not available: see fulltext.], there exists a linear subspace [InlineMediaObject not available: see fulltext.] 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 ≠ x ∈ X there exists some k ∈ [InlineMediaObject not available: see fulltext.] such that every null space containing x has dimension at most k. In particular, P has no infinite dimensional null space. For a given n odd and a cardinal τ , we obtain a cardinal N = N(τ, n) = exp n +1 τ such that every n-homogeneous polynomial on a real Banach space X of density N has a null space of density τ .
Positivity – Springer Journals
Published: Oct 13, 2006
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