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?](https://mathoverflow.net/questions/20946/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.



Determine whether a system of polynomials with real coefficients has a real solution?
The system can be put in the form $Y=LX-G(X)$. Here $X$ is the column vector with the unknowns $X\_1,\ldots,X\_s$. $Y$ is the column vector with components $Y\_i=-a\_{0,\ldots,0}^{(i)}$. $L$ is the matrix describing the linear part of your system, namely,
$$
L\_{ij}=a\_{0,\ldots,0,1,0,\ldots,0}^{(i)}
$$
with the $1$ at the $j$-th position. Finally
$$
G\_i(X)=-\sum\_{j\_1,\ldots,j\_s} a\_{j\_1,\ldots,j\_s}^{(i)}
X\_1^{j\_1}\cdots X\_s^{j\_s}
$$
where you only keep terms of degree at least 2.


Then your criterion and the Gershgorin circle theorem guarantee that the matrix $L$ is invertible so you can try to solve iteratively
$$
X=L^{-1}Y+L^{-1}G(X)
$$ 
$$
X=L^{-1}Y+L^{-1}G(L^{-1}Y+L^{-1}G(X))
$$
etc.
In the fact one can write the end result as an explicit series in terms of trees.
See for instance Theorem 1 in my article ["The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory"](http://link.springer.com/article/10.1007/s00023-003-0127-7). It is possible to write explicit bounds which imply the convergence of this series and therefore the existence of a real solution. Such bounds essentially mean that $Y$ and $G$ are sufficiently small. 
The question is whether your criterion implies such bounds. In the above paper the particular case treated is that where $G$ is homogeneous. To see how the formulas look like in the nonhomogenous case you can see my other article ["Feynman Diagrams in Algebraic Combinatorics"](http://www.emis.de/journals/SLC/wpapers/s49abdess.html).


