On the positivity of symmetric polynomial functions.

Timofte, Vlad

doi:10.1016/s0022-247x(03)00301-9

Cited by 52 publications

(28 citation statements)

References 14 publications

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“…If the witness set S is sufficiently simple then verifying that F is nonnegative becomes a simpler problem. The work of several authors has provided important and interesting examples of nonnegativity witness sets for symmetric polynomials [2], [4], [16], [11], [6]. The following theorems are some representative results, Theorem (Choi-Lam-Reznick [2]).…”

Section: Introductionmentioning

confidence: 99%

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Test sets for nonnegativity of polynomials invariant under a finite reflection group

Velasco¹

2016

Journal of Pure and Applied Algebra

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Abstract. A set S ⊂ R n is a nonnegativity witness for a set U of real homogeneous polynomials if F in U is nonnegative on R n if and only if it is nonnegative at all points of S. We prove that the union of the hyperplanes perpendicular to the elements of a root system Φ ⊆ R n is a witness set for nonnegativity of forms of low degree which are invariant under the reflection group defined by Φ. We prove that our bound for the degree is sharp for all reflection groups which contain multiplication by −1. We then characterize subspaces of forms of arbitrarily high degree where this union of hyperplanes is a nonnegativity witness set. Finally we propose a conjectural generalization of Timofte's half-degree principle for finite reflection groups. IntroductionThe general problem of deciding whether a homogeneous polynomial F in n variables with real coefficients is nonnegative has been one of the guiding questions of real algebraic geometry since the time of Hilbert. The intrinsic interest of this question has been complemented by a recent surge in its applications to polynomial optimization [1], control theory [9] and to the general theory of aproximation algorithms [14] among others. For all these applications, finding simple and efficient certificates for guaranteeing the nonnegativity of a given polynomial is of paramount importance.In the case of polynomials which are invariant under symmetric groups, such certificates have taken the form of "nonnegativity witness sets", that is subsets S ⊂ R n such that nonnegativity of F on S is equivalent to the nonnegativity of F in R n . If the witness set S is sufficiently simple then verifying that F is nonnegative becomes a simpler problem. ). An n-ary even symmetric sextic is nonnegative if and only if it is so at the n points (1, 0 . . . , 0), (1, 1, 0, . . . , 0), . . . , (1, 1 . . . , 1).2010 Mathematics Subject Classification. Primary 14P05 Secondary 20F55.

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Section: Introductionmentioning

confidence: 99%

“…Theorem (Timofte's half-degree principle [16], [11]). An n-variate symmetric polynomial of degree 2d is nonnegative if and only if it is so at every point with at most max{2, d} distinct components.…”

Section: Introductionmentioning

confidence: 99%

Test sets for nonnegativity of polynomials invariant under a finite reflection group

Velasco¹

2016

Journal of Pure and Applied Algebra

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“…It must be pointed out that Theorem 6.2 can be operated artificially. Theorem 6.2 is different from the result in [15], because that of [15] only has meaning for n ≥ [d/2] (i.e., the greatest integer function of d/2) and can be operated artificially. The problem in Example 6.3 is too difficult, and furthermore, it cannot be solved by all the softwares in the existing circumstances.…”

Section: The Sufficient Condition That Inequality (16) Holdsmentioning

confidence: 94%

“…Our methods are, of late years, the approach of descending dimension and theory of majorization; and apply some techniques of mathematical analysis and permanents [12] in algebra. Note that the way of descending dimension used in this paper is different from [15,23,25]; and the majorization is an effective theory that "it can state the inwardness and the relation between the quantities" (see [4,11,16]). It is very interesting that the mathematical analysis and permanent can skillfully be combined.…”

Section: Definition 12mentioning

confidence: 99%

The optimization for the inequalities of power means

Wen

Wang

2006

Journal of Inequalities and Applications

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Let M [t] n (a) be the tth power mean of a sequence a of positive real numbers, where a = (a 1 ,a 2 ,...,a n ),n ≥ 2, and α,λ ∈ R m ++ ,m ≥ 2, m j=1 λ j = 1,min{α} ≤ θ ≤ max {α}. In this paper, we will state the important background and meaning of the inequalityn (a); a necessary and sufficient condition and another interesting sufficient condition that the foregoing inequality holds are obtained; an open problem posed by Wang et al. in 2004 is solved and generalized; a rulable criterion of the semipositivity of homogeneous symmetrical polynomial is also obtained. Our methods used are the procedure of descending dimension and theory of majorization; and apply techniques of mathematical analysis and permanents in algebra.Copyright © 2006 J. Wen and W.-L. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Symbols and introductionWe will use some symbols in the well-known monographs [1,5,13]:is a finite set, and it is not empty.Recall that the definitions of the tth power mean and Hardy mean of order r for a sequence a = (a 1 ,...,a n ) (n ≥ 2) are, respectively,, if 0< |t| < +∞,n (a) = n √ a 1 a 2 ··· a n , if t = 0, Hindawi Publishing Corporation

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“…Intuitively speaking, v(x) counts the distinct components of x, while v * (x) counts the non-zero distinct components only. [11,12] . (…”

Section: Problem Description and Preliminariesmentioning

confidence: 99%

A class of mechanically decidable problems beyond Tarski’s model

2007

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By means of dimension-decreasing method and cell-decomposition, a practical algorithm is proposed to decide the positivity of a certain class of symmetric polynomials, the numbers of whose elements are variable. This is a class of mechanically decidable problems beyond Tarski model. To implement the algorithm, a program nprove written in maple is developed which can decide the positivity of these polynomials rapidly.

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On the positivity of symmetric polynomial functions.

Abstract: We show that positivity on R n + and on R n of real symmetric polynomials of degree at most p in n ≥ 2 variables is solvable by algorithms running in poly(n) time. For real symmetric quartics, we find explicit discriminants and related Maple algorithms running in lin(n) time.

Cited by 52 publications

References 14 publications

Test sets for nonnegativity of polynomials invariant under a finite reflection group

Test sets for nonnegativity of polynomials invariant under a finite reflection group

The optimization for the inequalities of power means

A class of mechanically decidable problems beyond Tarski’s model

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