Odd degree polynomials on real Banach spaces
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. Using the Erdős-Rado theorem, the authors prove that increasing the density of \(X\) leads to the increase of the density of the null spaces. A space \(X\) is said to have density \(\Gamma\) if there exists a set \(S\subset X\) of cardinality \(\Gamma\) such that \(\overline{S}=X\), and, moreover, \(\Gamma\) is the minimal cardinal with this property. To be more precise, denote by \(\exp\alpha= 2^\alpha\) and \(\exp^{n+1}\alpha=\exp(\exp \alpha)\). Assuming the generalized continuum hypothesis (GCH), it is proved that, if card\((\Gamma) \geq \exp^n \alpha\), \(\alpha \geq c\) and \(n\) is odd, then every \(n\)-homogeneous polynomial on \(\ell_1(\Gamma)\) has a null space of density at least \(\alpha^+\). Again assuming the GCH, it is proved that for any real Banach space \(X\) of density at least \(\exp^{n+1} \alpha\) with \(\alpha \geq c\) and any odd integer \(n\), every \(n\)-homogeneous polynomial on \(X\) has a null space of desity al leat \(\alpha^+\). Finally, some results and examples related to \(\ell_p(\Gamma)\) spaces are presented.
scientific article; zbMATH DE number 5156948