Fix $k,n$.
I want to describe the following set:
$\mathbb{P}^* =\{P\;\text{is a homogeneous polynomial of degree}\;k\;\text{with at least one negative coefficient}\;: P(x) \geq 0 \;\forall\;x \in \mathbb{R}^n_{+}\}$
I have proceeded as follows:
Any polynomial $P$ of the form below is in $\mathbb{P}^*$:
$P(x)=(a_{1,1}x_1+\cdots+a_{n,1}x_n)\cdots(a_{1,k-2}x_1+\cdots+a_{n,k-2}x_n)(a_{1,k-1}x_1+\cdots+a_{n,k-1}x_n)^2$
where $a_{1,1},\cdots,a_{n,1},\cdots,a_{1,k-2},\cdots,a_{n,k-2} \geq 0,$ with $a_{ij} > 0$ for at least one $i \in [n]$ for all $j \in [k-2]$, and at least one $i_1 \in [n]$ and one $i_2 \in [n]$ such that $a_{i_1,k-1} > 0$ and $a_{i_2,k-1} < 0$.
But this is obviously far from the whole set $\mathbb{P}^*$ because first of all depending on $k$ you can have more square terms etc., all of $a_{1,1},\cdots,a_{n,1},\cdots,a_{1,k-2},\cdots,a_{n,k-2}$ being non-negative is not necessary, and secondly this only describes polynomials with all real roots.
Thank you for your help.