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.

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.

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? Thank you for you help

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.

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? Thank you for you help

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? Thank you for you help