Writing Positive Polynomials as Sums of (Few) Squares
نویسندگان
چکیده
منابع مشابه
Approximating Positive Polynomials Using Sums of Squares
XS f dμ. It is natural to ask if the same is true for any linear functional L : R[X ] → R which is non-negative on MS. This is the Moment Problem for the quadratic module MS. The most interesting case seems to be when S is finite. A sufficient condition for it to be true is that each f ∈ T̃S can be approximated by elements of MS in the sense that there exists an element q ∈ R[X ] such that, for ...
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If a real polynomial f can be written as a sum of squares of real polynomials, then clearly f is nonnegative on R, and an explicit expression of f as a sum of squares is a certificate of positivity for f . This idea, and generalizations of it, underlie a large body of theoretical and computational results concerning positive polynomials and sums of squares. In this survey article, we review the...
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Acknowledgements A lot of people helped me during the work on this thesis. First of all, I am greatly indebted to Alexander Prestel, Claus Scheiderer and Markus Schweighofer. I benefitted a lot from their constant logistic and mathematical support. I also want to thank Robert Denk, David Grimm and Daniel Plaumann for many interesting discussions on the topic of this work. In 2007, I enjoyed a s...
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Hilbert's 17th problem concerns expressing polynomials on R as a sum of squares. It is well known that many positive polynomials are not sums of squares; see [R00] [deA preprt] for excellent surveys. In this paper we consider symmetric non-commutative polynomials and call one \matrix positive", if whenever matrices of any size are substituted for the variables in the polynomial the matrix value...
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We show that if a polynomial f ∈ R[x1, . . . , xn] is nonnegative on a closed basic semialgebraic set X = {x ∈ Rn : g1(x) ≥ 0, . . . , gr(x) ≥ 0}, where g1, . . . , gr ∈ R[x1, . . . , xn], then f can be approximated uniformly on compact sets by polynomials of the form σ0 + φ(g1)g1 + · · ·+ φ(gr)gr, where σ0 ∈ R[x1, . . . , xn] and φ ∈ R[t] are sums of squares of polynomials. In particular, if X...
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ژورنال
عنوان ژورنال: EMS Newsletter
سال: 2017
ISSN: 1027-488X
DOI: 10.4171/news/105/4