## There are significantly more nonnegative polynomials than sums of squares, arXiv preprint math.AG/0309130 (2003)

Citations: | 31 - 5 self |

### BibTeX

@MISC{Blekherman03thereare,

author = {Grigoriy Blekherman},

title = {There are significantly more nonnegative polynomials than sums of squares, arXiv preprint math.AG/0309130},

year = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

We investigate the quantitative relationship between nonnegative polynomials and sums of squares of polynomials. We show that if the degree is fixed and the number of variables grows then there are significantly more nonnegative polynomials than sums of squares. More specifically, we take compact bases of the cone of nonnegative polynomials and the cone of sums of squares and derive bounds for the volumes of the bases. If the degree is greater than 2 then we show that the ratio of the volumes of the bases, raised to the power reciprocal to the ambient dimension, tends to 0 as the number of variables tends to infinity. 1

### Citations

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(Show Context)
Citation Context ... there exists a form qv ∈ M such that Rewriting (3.1.2) we see that ( ∫ A ≤ λv(f) = 〈qv , f〉. SM 〈f , qv〉 2n ) 1 2n dµ . (3.1.3) There are explicit descriptions of the polynomials qv, see for example =-=[11]-=-, we will only need the property that for v ∈ S n−1 || qv||2 = √ DM. 6This can also be deduced by abstract representation theoretic considerations. We observe that ∫ 〈f , qv〉 SM 2n 1 n Γ(n + ) Γ(1 2 ... |

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Citation Context ...rst such explicit polynomials were constructed only fifty years later by Motzkin in the 1940’s. There are currently several known families of non-negative polynomials that are not sums of squares [3],=-=[9]-=-. 1There remains however a natural question, which we call, the Quantitative Sums of Squares Problem: Are there significantly more nonnegative polynomials than sums of squares of polynomials? The kno... |

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Citation Context ...squares; in all other cases there exist nonnegative polynomials that are not sums of squares. Hilbert’s proof of existence of nonnegative polynomials that are not sums of squares was non-constructive =-=[6]-=-. The first such explicit polynomials were constructed only fifty years later by Motzkin in the 1940’s. There are currently several known families of non-negative polynomials that are not sums of squa... |

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Citation Context ...e first such explicit polynomials were constructed only fifty years later by Motzkin in the 1940’s. There are currently several known families of non-negative polynomials that are not sums of squares =-=[3]-=-,[9]. 1There remains however a natural question, which we call, the Quantitative Sums of Squares Problem: Are there significantly more nonnegative polynomials than sums of squares of polynomials? The... |

9 |
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(Show Context)
Citation Context ...and let µ be the rotation invariant probability measure on SM. Then the following inequality for the average L∞ norm over SM holds: ∫ ||f||∞ dµ ≤ 2 √ 2n(2k + 1). SM Proof. It was shown by Barvinok in =-=[1]-=- that for all f ∈ Pn,2k, ( ) 1 2n 2kn + n − 1 ||f||∞ ≤ ||f||2n. 2kn By applying Stirling’s formula we can easily obtain the bound ( ) 1 2kn + n − 1 2kn 2n ≤ 2 √ 2k + 1. 5Therefore it suffices to esti... |

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Citation Context ... into M. Therefore we have ∫ 〈f , g 2 〉 2Dn,k ∫ 2 2Dn,k dµ(f) ≤ ||g || 2 〈f , p〉 2Dn,k dµ(f) for any p ∈ SM. SM We observe that SM ||g 2 ||2 = (||g||4) 2 and ||g||2 = 1. By a result of Duoandikoetxea =-=[4]-=- Corollary 3 it follows that ||g 2 ||2 ≤ 4 2k . 9Hence we obtain ∫ 〈f , g 2 〉 2Dn,k ∫ 4kDn,k dµ(f) ≤ 4 SM SV 〈f , p〉 2Dn,k dµ(f). We note that this bound is independent of g and substituting into (4.... |

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Citation Context ...roblem also has significance from the point of view of computational complexity. Using the tools of semidefinite programming one can efficiently compute whether a given polynomial is a sum of squares =-=[7]-=-. However, determining whether a polynomial is nonnegative is NP-hard for k ≥ 2 [2, Part 1]. Therefore we can ask how much we lose by testing for sums of squares instead of nonnegativity. There has be... |