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  • Definition 1. If a multivariate polynomial $f$ can be written as a finite sum of squared polynomials, i.e., $f(x)=\sum_{i = 1}^n g_i^2(x)$, then $f$ is SOS.

  • Definition 2. If an $n$-variate polynomial $f$ is nonnegative on $\Bbb R^n$, then $f$ is positive semidefinite (PSD).

We know all SOS polynomials are PSD. However, It is not correct that all PSD polynomials are SOS. For example, the Motzkin polynomial:

$$f(x,y)= 1 + x^2 y^4 + x^4 y^2 - 3 x^2 y^2 $$

Suppose $f(x_1, \dots ,x_n)$ is SOS and $\min\limits_{x \in \Bbb R^n} f =: r $. Is $f(x)-\alpha$ SOS for all $0 \le \alpha <r$?

If not, would you please give me a counterexample?

Edit

One way to give a counterexample is to consider a PSD polynomial $p(x)$ which is not SOS. Then, if $p(x) +k$ becomes SOS for $k=k^*$, $f(x)=p(x) +k^*$ can be our counterexample. This is because $f$ is SOS while $f-k$ is not for some $k\le k^*$.

$m(x,y)= 1 + x^2 y^4 + x^4 y^2 - 3 x^2 y^2 $ is not SOS but PSD. Also, according to https://arxiv.org/abs/math/0103170, $m(x,y)+k$ is not SOS for any real $k$. Unfortunately, this example neither proves my first question nor gives a counterexample.

khashayar
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  • You may want to take a look at Parrilo & Sturmfels's Minimizing Polynomial Functions (2001) – Rodrigo de Azevedo Jun 02 '22 at 13:55
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    @RodrigodeAzevedo Indedd, in page 5 of the pdf you linked to it is stated that $f(x,y)-\lambda$ is not a sum of squares for any $\lambda\in{\mathbb R}$. – Ewan Delanoy Jun 05 '22 at 09:14
  • @EwanDelanoy Good find. You may want to post an answer citing that paper. – Rodrigo de Azevedo Jun 05 '22 at 09:19
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    @RodrigodeAzevedo Actually, the OP is not quite clear on this, as it first talks about a particular $f$ (the Motzkin polynomial) and then proceeds to ask a question about a generic $f$ which is assumed to be SOS (which is not the case the Motzkin polynomial). So it seems that my comment does not answer the main question in the OP. Hopefully our asker friend will clarify this. – Ewan Delanoy Jun 05 '22 at 09:23

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