Today my students ask me following problem,

Define polynomial $P$ with real coefficients $$P_{j}=\sum_{i_{1},\cdots,i_{s}}a^{(j)}_{i_{1}\cdots i_{s}}X^{i_{1}}_{1}\cdots X^{i_{s}}_{s},(1\le j\le s)$$ where $s$ be give positive integer.

and Assmue that $a^{(1)}_{1,0,0,\cdots,0}a^{(2)}_{0,1,0,\cdots,0}\cdots a^{(s)}_{0,0,\cdots,1}$ is negatives,and such $$2\min\{|a^{(1)}_{1,0,\cdots,0}|,|a^{(2)}_{0,1,\cdots,0}|,\cdots,|a^{(s)}_{0,0,\cdots,1}|\}>\sum_{1\le j\le s,i_{1},i_{2},\cdots,i_{s}}|a^{(j)}_{i_{1},\cdots,i_{s}}|$$

show that: this System of polynomial equations $$P_{j}(X_{1},X_{2},\cdots,X_{s})=0,(1\le j\le s)$$ has solution $(X_{1},X_{2},\cdots,X_{S})\in R^{s}$

I know this reslut: Criteria to determine whether a real-coefficient polynomial has real root?

But for system of real coefficients have real roots,I can't find it any paper,maybe this reslut is old?