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|Title:||Effective Pólya semi-positivity for non-negative polynomials on the simplex|
|Citation:||Mok, H.-N., To, W.-K. (2008-08). Effective Pólya semi-positivity for non-negative polynomials on the simplex. Journal of Complexity 24 (4) : 524-544. ScholarBank@NUS Repository. https://doi.org/10.1016/j.jco.2008.01.003|
|Abstract:||We consider homogeneous polynomials f ∈ R [x1, ..., xn] which are non-negative on the standard simplex in Rn, and we obtain sufficient conditions for such an f to be Pólya semi-positive, that is, all the coefficients of (x1 + ⋯ + xn)N f are non-negative for all sufficiently large positive integers N. Such sufficient conditions are expressed in terms of the vanishing orders of the monomial terms of f along the faces of the simplex. Our result also gives effective estimates on N under such conditions. Moreover, we also show that any Pólya semi-positive polynomial necessarily satisfies a slightly weaker condition. In particular, our results lead to a simple characterization of the Pólya semi-positive polynomials in the low dimensional case when n ≤ 3 as well as the case (in any dimension) when the zero set of the polynomial in the simplex consists of a finite number of points. We also discuss an application to the representations of non-homogeneous polynomials which are non-negative on a general simplex. © 2008 Elsevier Inc. All rights reserved.|
|Source Title:||Journal of Complexity|
|Appears in Collections:||Staff Publications|
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