Odd degree polynomials on real Banach spaces
type: Item
instance of: scholarly article
title: Odd degree polynomials on real Banach spaces
zbMATH Open document ID: 1158.46030
author: Petr Hájek
author: Richard M. Aron
published in: Positivity
publication date: 2007-05-24T00:00:00Z
review text: A classical theorem of \textit{B. J. Birch} [Mathematika, Lond. 4, 102--105 (1957; Zbl 0081.04501)] states that for any positive integers \(n\) and \(k\), \(n\) odd, there exists a finite dimensional space of high enough dimension \(N(n,k)\) such that every \(n\)-homogeneous polynomial vanishes on a subspace of dimension at least \(k\). For complex infinite dimensional separable spaces, \textit{A. Plichko} and \textit{A. Zagorodnyuk} [J. Math. Anal. Appl. 220, No.~2, 477--494 (1998; Zbl 0919.46036)] showed that any complex polynomial has an infinite dimensional null space. The real scalar version of this result fails, since such spaces admit a positive (away from the origin) \(2k\)-homogeneous polynomial for every \(k\). In the paper under review, the authors investigate the size of the null set for any odd degree homogeneous polynomial defined on an infinite dimensional real Banach space. They first study the case when the underlying space \(X\) is separable (or, more generally, a space with \(w^*\)-separable dual \(X^*\)). They show that it is always possible to find an \(n\)-homogeneous polynomial \(P: X\to\mathbb R\) without an infinite null space, for any odd integer \(n>1\). More precisely, the authors show that given any \(x\in X\), \(x\neq 0\), with \(P(x)=0\), there exists \(N\in \mathbb N\) such that every null space containing \(x\) has dimension at most \(N\). In order to prove their result, they construct a 3-homogeneous polynomial on \(c_0\) satisfying the hypothesis of the statement and show how to produce the desired \(n\)-homogeneous polynomial on \(X\) from that on \(c_0\). This result applies to all separable Banach spaces \(\ell_\infty\) and \(C(K)\), where \(K\) is separable (not necessarily metrizable). Since it is inherited by subspaces, it also applies to \(\ell_1(c)\). The second part of the paper is devoted to investigating the size of maximal null sets in the nonseparable setting, where the spaces \(\ell_1(\Gamma)\) play a significant role…
reviewed by: Silvia Lassalle
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46B26
zbMATH DE Number: 5156948
zbMATH Keywords: odd degree polynomials
zbMATH Keywords: real Banach spaces
zbMATH Keywords: zero sets
MaRDI profile type: Publication
full work available at URL: s11117-006-2035-9
OpenAlex ID: W2071526738
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DOI: 10.1007/S11117-006-2035-9
description: scientific article; zbMATH DE number 5156948