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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>>