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I encountered a polynomial (homogeneous quartic form) in 8 variables that numerics show to be non-negative but also an optimization procedure shows that a sum-of-squares decomposition does not (apparently) exist. The polynomial is:

$$ff = (a_1^2 + a_2^2 + a_3^2 + a_4^2)^2 - 2 (a_2 b_2 - a_3 b_3)^2 - (a_3 b_1 + a_4 b_2 + a_1 b_3 + a_2 b_4)^2 \\ - (a_2 b_1 + a_1 b_2 + a_4 b_3 + a_3 b_4)^2 - 2 (a_1 b_1 - a_4 b_4)^2 + (b_1^2 + b_2^2 + b_3^2 + b_4^2)^2 $$

I'm wondering what algebraic tools would allow me to prove non-negativity of this polynomial if the SOS approach fails to work.

1 Answers 1

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It appears to be SOS-decomposible. Easily solved using the sum-of-squares module in YALMIP, with Mosek as SDP solver. By symmetry it also decomposes to a sum of smaller sum-of-squares

sdpvar a1 a2 a3 a4
sdpvar b1 b2 b3 b4

ff = (a1^2 + a2^2 + a3^2 + a4^2)^2 - 2*(a2*b2 - a3*b3)^2 - (a3*b1 + a4*b2 +    a1*b3 + a2*b4)^2 - (a2*b1 + a1*b2 + a4*b3 + a3*b4)^2 - 2*(a1*b1 - a4*b4)^2 + (b1^2 + b2^2 + b3^2 + b4^2)^2

[~,v,Q] = solvesos(sos(ff))];

Post-processing by studying the Gramian matrix $Q$ shows that there is more to be done, and that you most likely can find a simple certfied analytical expression.