18,658 research outputs found
Norms as products of linear polynomials
Let F be a number field, and let F\subset K be a field extension of degree n.
Suppose that we are given 2r sufficiently general linear polynomials in r
variables over F. Let X be the variety over F such that the F-points of X
bijectively correspond to the representations of the product of these
polynomials by a norm from K to F. Combining the circle method with descent we
prove that the Brauer-Manin obstruction is the only obstruction to the Hasse
principle and weak approximation on any smooth and projective model of X.Comment: 25 page
The symmetric Radon-Nikod\'ym property for tensor norms
We introduce the symmetric-Radon-Nikod\'ym property (sRN property) for
finitely generated s-tensor norms of order and prove a Lewis type
theorem for s-tensor norms with this property. As a consequence, if is
a projective s-tensor norm with the sRN property, then for every Asplund space
, the canonical map is a metric surjection. This
can be rephrased as the isometric isomorphism for certain polynomial ideal \Q. We also relate the sRN
property of an s-tensor norm with the Asplund or Radon-Nikod\'{y}m properties
of different tensor products. Similar results for full tensor products are also
given. As an application, results concerning the ideal of -homogeneous
extendible polynomials are obtained, as well as a new proof of the well known
isometric isomorphism between nuclear and integral polynomials on Asplund
spaces.Comment: 17 page
Unconditionality in tensor products and ideals of polynomials, multilinear forms and operators
We study tensor norms that destroy unconditionality in the following sense:
for every Banach space with unconditional basis, the -fold tensor
product of (with the corresponding tensor norm) does not have unconditional
basis. We establish an easy criterion to check weather a tensor norm destroys
unconditionality or not. Using this test we get that all injective and
projective tensor norms different from and destroy
unconditionality, both in full and symmetric tensor products. We present
applications to polynomial ideals: we show that many usual polynomial ideals
never enjoy the Gordon-Lewis property. We also consider the unconditionality of
the monomial basic sequence. Analogous problems for multilinear and operator
ideals are addressed.Comment: 23 page
Geometry of integral polynomials, -ideals and unique norm preserving extensions
We use the Aron-Berner extension to prove that the set of extreme points of
the unit ball of the space of integral polynomials over a real Banach space
is . With this description we show
that, for real Banach spaces and , if is a non trivial -ideal in
, then (the -th symmetric
tensor product of endowed with the injective symmetric tensor norm) is
\emph{never} an -ideal in . This
result marks up a difference with the behavior of non-symmetric tensors since,
when is an -ideal in , it is known that
(the -th tensor product of endowed
with the injective tensor norm) is an -ideal in
. Nevertheless, if is Asplund, we prove
that every integral -homogeneous polynomial in has a unique extension to
that preserves the integral norm. We explicitly describe this extension.
We also give necessary and sufficient conditions (related with the continuity
of the Aron-Berner extension morphism) for a fixed -homogeneous polynomial
belonging to a maximal polynomial ideal \Q(^kX) to have a unique norm
preserving extension to \Q(^kX^{**}). To this end, we study the relationship
between the bidual of the symmetric tensor product of a Banach space and the
symmetric tensor product of its bidual and show (in the presence of the BAP)
that both spaces have `the same local structure'. Other applications to the
metric and isomorphic theory of symmetric tensor products and polynomial ideals
are also given.Comment: 25 page
Non-linear Plank Problems and polynomial inequalities
We study lower bounds for the norm of the product of polynomials and their
applications to the so called \emph{plank problem.} We are particularly
interested in polynomials on finite dimensional Banach spaces, in which case
our results improve previous works when the number of polynomials is large.Comment: 19 page
Polynomial Norms
In this paper, we study polynomial norms, i.e. norms that are the
root of a degree- homogeneous polynomial . We first show
that a necessary and sufficient condition for to be a norm is for
to be strictly convex, or equivalently, convex and positive definite. Though
not all norms come from roots of polynomials, we prove that any
norm can be approximated arbitrarily well by a polynomial norm. We then
investigate the computational problem of testing whether a form gives a
polynomial norm. We show that this problem is strongly NP-hard already when the
degree of the form is 4, but can always be answered by testing feasibility of a
semidefinite program (of possibly large size). We further study the problem of
optimizing over the set of polynomial norms using semidefinite programming. To
do this, we introduce the notion of r-sos-convexity and extend a result of
Reznick on sum of squares representation of positive definite forms to positive
definite biforms. We conclude with some applications of polynomial norms to
statistics and dynamical systems
A Lindenstrauss theorem for some classes of multilinear mappings
Under some natural hypotheses, we show that if a multilinear mapping belongs
to some Banach multlinear ideal, then it can be approximated by multilinear
mappings belonging to the same ideal whose Arens extensions simultaneously
attain their norms. We also consider the class of symmetric multilinear
mappings.Comment: 11 page
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