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I have read that it remains an open question, whether an example can be constructed of a non-negative convex polynomial that cannot be written as a sum-of-squares. My reading includes the following post: Polynomials that are sums of squares

However, the material I have found is generally not so recent, and so I simply wondered whether any examples have been discovered in more recent years?

Thank you.

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    $\begingroup$ I guess you are implicitly also referring to arxiv.org/abs/0910.0656 for an existence result...? $\endgroup$
    – Suvrit
    Commented Nov 17, 2020 at 0:31
  • $\begingroup$ Sorry my post was vague in this regard. That is a useful reference, which "implies existence of convex forms that are not sums of squares, although there are still no known examples." As a reference from 2009, I wonder whether this still largely represents where we are today? $\endgroup$
    – Brian
    Commented Nov 17, 2020 at 14:45
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    $\begingroup$ Brian, are you acquainted with the following? Amir Ali Ahmadi, Alex Olshevsky, Pablo A. Parrilo, John N. Tsitsiklis, NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems, December 2010. $\endgroup$
    – Rodrigo de Azevedo
    Commented Nov 17, 2020 at 15:33
  • $\begingroup$ I have read a really interesting paper by R. Bielawski, which constructs non-negative polynomials which are not sums of squares. But he does not mention convexity. I just mention it in case it could be helpful. $\endgroup$
    – Malkoun
    Commented Nov 17, 2020 at 17:41
  • $\begingroup$ Thank you Malkoun. It does seem that finding examples in the convex case is much harder as there do not seem to be any so far. $\endgroup$
    – Brian
    Commented Nov 17, 2020 at 17:49

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