# Every non-negative multivariate polynomial has degree even?

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.

• Yes, it is true. Mar 24 '18 at 19:18

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