Carly Cannon

2022-07-08

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 $j\in [k]$?

Zichetti4b

Beginner2022-07-09Added 13 answers

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 $P(u)=0$ for all $u\in {B}_{1}\times {B}_{2}\times \dots \times {B}_{n}$ with all ${B}_{i}$ infinite.

dream13rxs

Beginner2022-07-10Added 4 answers

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