I'm seeking for a *Certificate of Positivity* for the AM-GM inequality in five variables
$$a^5+b^5+c^5+d^5+e^5-5abcde\;\ge 0\qquad\forall\,a,b,c,d,e\ge 0\,.$$



> 
> Can one write the LHS as a sum
>  $\,\sum\_i h\_i\,s\_i\,$ with real polynomials
>  $\,h\_i(a,b,c,d,e)\,$ and $\,s\_i(a,b,c,d,e)$, where
> 
> 
> * each $\,h\_i\,$ is homogeneous of degree $1$ and positive (with arguments $\ge0\,$),
> * each $\,s\_i\,$ is a square?
> 
> 
> 


In the case of $3$ variables the answer would be *yes* by the common factorisation
$$a^3+b^3+c^3-3abc\;=\;\frac 12(a+b+c)\left[(a-b)^2+(b-c)^2+(c-a)^2\right].$$


This is a [Cross-post](https://math.stackexchange.com/q/2397904/316749)
from math.SE after a decent period of waiting ...


Remark: From David's comment to
[this post](https://math.stackexchange.com/q/2244565/316749)
the $n=5$ expression does not factor according to Maple, contrary to the preceding $\,n=3\,$ case.




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*Added in edit:*  

I am really delighted by the community's rich spectrum of reactions, such a Math Overflow within the 12 hours after posting!  

*Thanks a lot!*


In particular I've gotten a more general answer than hoped for, covering the specific issue addressed.
If you'd like to see a specific five-variables-certificate as initially sought-after, then you may follow the above "Cross-post" link, where a corresponding answer has been added.



Wanted: Positivity certificate for the AM-GM inequality in low dimension
The following paper:



> 
> Fujiwara, Kazumasa, and Tohru Ozawa. [Identities for the Difference between the Arithmetic and Geometric Means,](http://m-hikari.com/ijma/ijma-2014/ijma-29-32-2014/ozawaIJMA29-32-2014.pdf) (2014). 
> 
> 
> 


proves the following representation for **odd $n$:**


\begin{equation\*}
\frac{1}{n}\sum\_i x\_i^n - \prod\_i x\_i = \sum\_{i=1}^n x\_i\sum\_{j \in J(n)} (P\_{ij}(x\_1,\ldots,x\_n))^2,
\end{equation\*}
for suitable polynomials $P\_{ij}$. For **even $n$**, a SOS representation is available in Ch.2 of Hardy, Littlewood, Polyá.


