Abstract
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 , there exists a linear subspace 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 ∈ 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) = expn+1τ such that every n-homogeneous polynomial on a real Banach space X of density N has a null space of density τ .
Similar content being viewed by others
References
R. Aron, R. Gonzalo, A. Zagorodnyuk, Zeroes of real polynomials, Linear Multilinear Algebra 48, No. 2, (2000), 107–115.
R. Aron, P. Hájek, Zero sets of polynomials in several variables, to appear in Archiv. der Math.
B.J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika 4 (1957), 102–105.
P. Erdös, A. Hajnal, A. Mate, R. Rado, Combinatorial set theory: partition relations for cardinals, Studies in Logic, vol. 106, North-Holland (1984).
M. Fabian, P. Habala, P. Hájek, V. Montesinos, J. Pelant, V. Zizler, Functional Analysis and Infinite Dimensional Geometry, Springer Verlag, CMS Books in Mathematics 8 (2001).
P. Habala, P. Hájek, Stabilization of polynomials, C. R. Acad. Sci. Paris 320, I (1995), 821–825.
P. Hájek, Polynomial algebras on classical Banach spacesi, Israel J. Math. 106 (1998), 209–220.
A. Plichko, A. Zagorodnyuk, On automatic continuity and three problems of ``The Scottish Book'' concerning the boundedness of polynomial functionals, J. Math. Anal. Appl. 220 (1998), 477–494.
W.M. Schmidt, The density of integer points on homogeneous varieties, Acta Math. 154 (1985), 243–296.
T.D. Wooley, Linear spaces of cubic hypersurfaces, and pairs of homogeneous cubic equations, Bull. London Math. Soc. 29 (1997), 556–562.
T.D. Wooley, An explicit version of Birch's theorem, Acta Arithm. 85 (1998), 79–96.
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
Aron, R., Hájek, P. Odd Degree Polynomials on Real Banach Spaces. Positivity 11, 143–153 (2007). https://doi.org/10.1007/s11117-006-2035-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-006-2035-9