Suppose there are $k$ non-zero homogeneous polynomials ${P}^{1}(.),\cdots ,{P}^{k}(.)$, each of degree r in n variables, such that ${P}^{j}({x}_{1},\cdots ,{x}_{n})\ge 0$ for all $({x}_{1},\cdots ,{x}_{n})\ge 0$, for all $j\in [k]$. Under what conditions (on the ${P}^{j}(.)$s) would there exist an $\alpha \in {\mathbb{R}}_{+}^{n}$ such that ${P}^{j}(\alpha )>0$ for all $j\in [k]$?