Today my students asked me the following problem：

Define polynomials $P_j$ with real coefficients $$P_{j}=\sum_{i_{1},\cdots,i_{s}}a^{(j)}_{i_{1}\cdots i_{s}}X^{i_{1}}_{1}\cdots X^{i_{s}}_{s}, \qquad (1\le j\le s)$$ where $s$ is a given positive integer.

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

Show that this system of polynomial equations $$P_{j}(X_{1},X_{2},\ldots,X_{s})=0, \qquad(1\le j\le s)$$ has a solution $(X_{1},X_{2},\ldots,X_{s})\in \mathbb{R}^{s}$

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

But for a system of real polynomials to have a real solution, I can't find any similar results in the literature. Maybe this result is old? Thank you for you help.