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Symmetric nonnegative forms and sums of squares. (arXiv:1205.3102v3 [math.OC] UPDATED)

来源于:arXiv
We study the relationship between symmetric nonnegative forms and symmetric sums of squares. Our particular emphasis is on the asymptotic behavior when the degree 2d is fixed and the number of variables $n$ grows. We show that in sharp contrast to the general case the difference between symmetric forms and sums of squares does not grow arbitrarily large for any fixed degree 2d. For degree 4 we show that the difference between symmetric nonnegative forms and sums of squares asymptotically goes to 0. More precisely we relate nonnegative symmetric forms to symmetric mean inequalities, valid independent of the number of variables. Given a symmetric quartic we show that the related symmetric mean inequality holds for all $n\geq 4$, if and only if the symmetric mean inequality can be written as a sum of squares. We conjecture that this is true for arbitrary degree 2d. 查看全文>>