Counterexample


There is an explicit counterexample to my proposal for polynomial complexity solution to max-cut problem (see The P-versus-NP page entry #63 ):

The quartic polynomial \sum_{i=1}^5 (x_i^2-1)^2 + ( \sum x_i )^2 -.1 is positive definite, e.g > 0 for all x, but cannot be represented as a sum of squares. On the other hand, this polynomial represent fully connected graph. The polynomial is due to Pablo Parrilo via Amir Ali Ahmadi.

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