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 from math.SE after a decent period of waiting ...
Remark: From David's comment to this post the $n=5$ expression does not factor according to Maple, contrary to the preceding $\,n=3\,$ case.
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.