I am interested in solutions of the following system of polynomial equations that can be written as an 'element-wise' vector polynomial:
$$ \mathbf{0} = \mathbf{A}^{(0)}\mathbf{1} + \mathbf{A}^{(1)}\mathbf{x} + \mathbf{A}^{(2)}\mathbf{x}^{\circ 2} + \dots + \mathbf{A}^{(k)}\mathbf{x}^{\circ k} = \sum_{n=0}^k \mathbf{A}^{(n)} \mathbf{x}^{\circ n}, $$
where $\mathbf{x} \in \mathbb{R}^m$, $\mathbf{A}^{(j)} \in \mathbb{R}^{m \times m}$, and $\mathbf{1}$ is a vector of ones. Importantly, $\mathbf{x}^{\circ n}$ is the element-wise (Hadamard) exponential, i.e., $(\mathbf{x}^{\circ n})_i = x_i^{n}$.
For $k=1$ this is a simple linear system. A solution for $k=2$ would already be interesting to me. In that case the system reads:
\begin{align} 0 =& a^{(0)}_{1} + a^{(1)}_{11} x_1 + \dots + a^{(1)}_{1m} x_m + a^{(2)}_{11} x_1^2 + \dots + a^{(2)}_{1m} x_m^2 \\ \vdots &\\ 0 =& a^{(0)}_{m} + a^{(1)}_{m1} x_1 + \dots + a^{(1)}_{mm} x_m + a^{(2)}_{m1} x_1^2 + \dots + a^{(2)}_{mm} x_m^2, \end{align}
or in vector notation with $ \mathbf{A} \in \mathbb{R}^{m \times\ m}, \mathbf{b} \in \mathbb{R}^m$:
$$ \mathbf{0} = \mathbf{x}^{\circ 2} + \mathbf{A}\mathbf{x} + \mathbf{b}. $$