It is wellknown that a positive univariate polynomial $p(x)>0$ for all $x\in R$, can be written as a sum of squares: $p(x) = \sum_{i=1}^n q_i^2(x)$, and I found references saying (without any details) that this is not true for multivariate polynomials, but intuitively I can not understand why. I am wondering if there are some counter examples, i.e. multivariate positive polynomials that can not be written as a sum of squares. thanks in advance.
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1 See this MathOverflow question and its answers. In particular, the Motzkin polynomial $1 + x^2 y^4 + x^4 y^2 - 3 x^2 y^2 $ is nonnegative but not a sum of squares of polynomials.
- $\begingroup$ the example is very helpful, thanks! $\endgroup$ljl– ljl2015-02-24 08:14:02 +00:00Commented Feb 24, 2015 at 8:14