Positivity 7: 285–295, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Zeros of Polynomials on Banach Spaces: The Real
and I. ZALDUENDO
Department of Mathematics, Kent State University, Kent, OH 44242, USA (Current address:
School of Mathematics, Trinity College, Dublin 2, Ireland e-mail: firstname.lastname@example.org);
Department of Mathematics, University College, Dublin 4, Ireland (e-mail: email@example.com);
Department of Mathematics, National University of Ireland, Galway, Ireland (e-mail:
Instituto Argentino de Matematico, CONICET, Saavedra 15–3en piso,
C1083ACA, Buenos Aires, Argentina (email: firstname.lastname@example.org)
Received 8 March 2001; accepted 11 February 2002
Abstract. Let E be a real Banach space. We show that either E admits a positive deﬁnite 2-
homogeneous polynomial or every 2-homogeneous polynomial on E has an inﬁnite dimensional
subspace on which it is identically zero. Under addition assumptions, we show that such subspaces
are non-separable. We examine analogous results for nuclear and absolutely (1,2)-summing 2-homo-
geneous polynomials and give necessary and sufﬁcient conditions on a compact set K so that C(K)
admits a positive deﬁnite 2-homogeneous polynomial or a positive deﬁnite nuclear 2-homogeneous
The study of the zeros of a complex polynomial has a long history, with results
coming via complex analysis, algebraic geometry, and functional analysis (see,
e.g., [8, 9, 12]). On the other hand, although similar studies for real polynomi-
als exist [4, 1], they seem somewhat less common. The case of the polynomial
P : R
→ R, P(x) =
notwithstanding, it is exactly the zeros of real
valued 2-homogeneous polynomials which will be of interest here, in the case
when the domain R
is replaced by an inﬁnite dimensional real Banach space E.
There are many ‘large’ Banach spaces E for which there is no positive deﬁnite
2-homogeneous polynomial P . As we will see, for a real Banach space E, either
E admits a positive deﬁnite 2-homogeneous polynomial or every 2-homogeneous
polynomial on E is identically zero on an inﬁnite dimensional subspace of E.
Part of this work was done while the author was a Fulbright visitor to the Universidad de San
es, Buenos Aires, Argentina, to whom he expresses his thanks.
Part of this work was done while the author was a European Union Postdoctoral Fellow at the
The author acknowledges the support of a Forbairt Basic Research Grant.
Part of this work was done while the author was visiting the NUI Galway.