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Is it true that every non-negative multivariate polynomial with $n$ variable on $\mathbb R$ has even degree?

By degree of polynomial I mean greatest sum of powers of variables for each monomial.

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    $\begingroup$ Yes, it is true. $\endgroup$ Mar 24 '18 at 19:18
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Indeed, suppose any variable $y$ appears to a greatest power $m$ which is odd (holding all other variables constant such that the coefficient of $y^m$ is nonzero). Then by sending $y$ to $-\infty$, we can make the expression become as negative as we wish.

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  • $\begingroup$ Perhaps we have many monomials which has $y^m$ and all have same degree! $\endgroup$
    – user522529
    Mar 24 '18 at 21:02
  • $\begingroup$ @Nothing Substitute constants for all the other variables, such that the remaining $y^m$ term is nonzero. $\endgroup$ Mar 24 '18 at 21:05
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    $\begingroup$ @PatrickStevens I don't think I fully understood your argument. How would you choose the variable $y$ if the polynomial is $p(x_1,x_2,x_3) = x_1 x_2 x_3 + x_1^2 + x_2^2 + x_3^2$? $\endgroup$
    – Yibo Yang
    May 14 '19 at 15:55

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