Positive Polynomials and Sequential Closures of Quadratic Modules

Details Positive Polynomials and Sequential Closures of Quadratic Modules Positive Polynomials and Sequential Closures of Quadratic Modules

2008-07-21

arxiv.org/abs/0807.3257v1

Mathematics

Algebraic Geometry

21 pages, 3 figures

Scientific paper

Let S be a basic closed semi-algebraic set in R^n and P the corresponding preordering in R[X_1,...,X_n]. We examine for which polynomials f there exist identities f+\ep q \in P for all \ep>0. These are precisely the elements of the sequential closure of P with respect to the finest locally convex topology. We solve the open problem whether this equals the double dual cone of P, by providing a counterexample. We then prove a theorem that allows to obtain identities for polynomials as above, by looking at a family of fibre-preorderings, constructed from bounded polynomials. These fibre-preorderings are easier to deal with than the original preordering in general. For a large class of examples we are thus able to show that either every polynomial f that is nonnegative on S admits such representations, or at least the polynomials from the double dual cone of P do. The results also hold in the more general setup of arbitrary commutative algebras and quadratic modules instead of preorderings.

Affiliated with

Also associated with

No associations

Rating

Positive Polynomials and Sequential Closures of Quadratic Modules does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Positive Polynomials and Sequential Closures of Quadratic Modules, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Positive Polynomials and Sequential Closures of Quadratic Modules will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-71975