# Suppose there are k non-zero homogeneous polynomials P 1 </msup> ( . ) ,

Suppose there are $k$ non-zero homogeneous polynomials ${P}^{1}\left(.\right),\cdots ,{P}^{k}\left(.\right)$, each of degree r in n variables, such that ${P}^{j}\left({x}_{1},\cdots ,{x}_{n}\right)\ge 0$ for all $\left({x}_{1},\cdots ,{x}_{n}\right)\ge 0$, for all $j\in \left[k\right]$. Under what conditions (on the ${P}^{j}\left(.\right)$s) would there exist an $\alpha \in {\mathbb{R}}_{+}^{n}$ such that ${P}^{j}\left(\alpha \right)>0$ for all $j\in \left[k\right]$?
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Zichetti4b
Let $B$ be an open ball with center ${x}_{0}$. Then $B$ contains ${x}_{0}+\sum _{i=1}^{n}{t}_{i}{e}_{i}$ for all i, where ${e}_{i}$ is the $i$-th unit vector and $\epsilon$ is a suitable positive number. Thus, if the polynomial $P$ vanish on $B$, es geht that $P\left(u\right)=0$ for all $u\in {B}_{1}×{B}_{2}×\dots ×{B}_{n}$ with all ${B}_{i}$ infinite.

dream13rxs
A nonzero polynomial of degree r has some rth partial derivative which is a nonzero constant. That's impossible for a function which is zero on an open set.