Authors: Richard M Aron Petr Hájek
Publish Date: 2006/10/13
Volume: 11, Issue: 1, Pages: 143-153
Abstract
A classical result of Birch claims that for given k n integers nodd there exists some N = Nk n such that for an arbitrary nhomogeneous polynomial P on Open image in new window there exists a linear subspace Open image in new window 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 wseparable dual X we construct an nhomogeneous polynomial P with the property that for every point 0 ≠ x ∈ X there exists some k ∈ Open image in new window 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 = Nτ n = exp n +1τ such that every nhomogeneous polynomial on a real Banach space X of density N has a null space of density τ Some of the work on this paper was done while the first author was a visitor to the Departamento de Análisis Matemático of the Universidad Complutense de Madrid to which great thanks are given The research of the second author was supported by grants Institutional Research Plan AV0Z10190503 A100190502 GA ČR 201/04/0090
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