This question was originally asked at stack exchange (https://math.stackexchange.com/questions/292922/multivariate-polynomial-with-all-coefficients-positive), but did not receive any feedback for more than a week. So I thought I’d ask it here too.

Let $n\geq 3$ be an integer. Consider the following polynomials :

$$ f(x_1,x_2, \ldots ,x_n)=\bigg(\frac{1}{n}\sum_{k=1}^n x_k^n\bigg)^{2n-2}- \bigg(\prod_{k=1}^n \frac{x_k^{2n-2}+\big(\prod_{j\neq k}x_j\big)^2}{2}\bigg) $$


$$ g(y_1,y_2,y_3, \ldots ,y_n)=f(y_1,y_1+y_2,y_1+y_2+y_3, \ldots, y_1+y_2+y_3+\ldots +y_n) $$

I conjecture (and I have checked with a computer that it is true for $n=3,4,5$) that

(1) On $(0,\infty)^n$, $g$ is increasing in each of its variables

(2) All the coefficients of $g$ are nonnegative.

Obviously, (2) is stronger (and probably harder to prove) than (1). Any ideas ? Any techniques known to deal with this kind of problem ?


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