Nonnegative functions as squares or sums of squares
نویسندگان
چکیده
منابع مشابه
Nonnegative Polynomials and Sums of Squares
A real polynomial in n variables is called nonnegative if it is greater than or equal to 0 at all points in R. It is a central question in real algebraic geometry whether a nonnegative polynomial can be written in a way that makes its nonnegativity apparent, i.e. as a sum of squares of polynomials (or more general objects). Algorithms to obtain such representations, when they are known, have ma...
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This is a translation of a paper [5] I wrote in 1971, and may help for Parimala’s course. Evidently completely outdated, but still may be useful. I changed some notation so as to be compatible with the course.
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In this paper we examine generalisations of the following problem posed by Laczkovich: Given an n × m rectangle with n and m integers, it can be written as a disjoint union of squares; what is the smallest number of squares that can be used? He also asked the corresponding higher dimensional analogue. For the two dimensional case Kenyon proved a tight logarithmic bound but left open the higher ...
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In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict positivity of a polynomial f on a basic closed set K ⊂ R. The interest in such identities originates not least from their importance in polynomial optimization. The majority of the important results requires the archimedean condition, which implies that K has to be compact....
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We study dimensions of the faces of the cone of nonnegative polynomials and the cone of sums of squares; we show that there are dimensional differences between corresponding faces of these cones. These dimensional gaps occur in all cases where there exist nonnegative polynomials that are not sums of squares. As either the degree or the number of variables grows the gaps become very large, asymp...
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ژورنال
عنوان ژورنال: Journal of Functional Analysis
سال: 2006
ISSN: 0022-1236
DOI: 10.1016/j.jfa.2005.06.011