Persistent failure of Euclidean quartic SOS exactness
An explicit rational homogeneous cubic with 25 monomials in 14 real variables has maximum one on the Euclidean unit sphere and admits a positive degree-four moment functional with value 500753357629/500000000000. For every positive real regularization parameter, quadratic strength and cubic strength, the associated quartic has an explicitly attained global minimum whose shift is not a sum of squares of real polynomials. A quantitative strict separation bounds every SOS lower bound. The polynomial is stationary at zero with negative definite Hessian. A second explicit 25-term cubic, certified…
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