= &k_{i1}s_1^2+k_{i2}s_2^2+k_{i3}s_3^2+k_{i4}s_1s_2+k_{i5}s_1s_3+\\
&k_{i6}s_2s_3+k_{i7}s_1+k_{i8}s_2+k_{i9}s_3+k_{i10}=0
\end{split}$$ where $k_{ij}$ ($ i \in \left[1,3\right], j \in \left[1,10\right] $) is an element of ${\bf{K}} = {\bf{C}}_2 - {\bf{A}}_2{\bf{A}}_1^{-1}{\bf{C}}_1$.

Hidden Variable Polynomial Solver {#subsec:hidden}
---------------------------------

Here we present a hidden variable method [@cox2006using] to solve (\[equ:29\]). We treat one unknown as a constant. This unknown is called the hidden variable. Without loss of generality, we treat $s_1$ and $s_2$ as unknowns, and $s_3$ as a constant. Thus (\[equ:29\]) can be rewritten as an equation system in $s_1$ and $s_2$ as $$\label{equ:30}
\begin{split}
{f_i}  = & {k_{i1}}s_1^2+{k_{i2}}s_2^2+{k_{i4}}{s_1}{s_2}+{p_{i1}}({s_3})s_1+ \\
&{p_{i2}}({s_3}){s_2}+{p_{i3}}({s_3}), i=1,2,3,
\end{split}$$ where ${p_{i1}}({s_3})={k_{i5}}{s_3}+{k_{i7}}$, ${p_{i2}}({s_3})={k_{i6}}{s_3}+{k_{i8}}$, ${p_{i3}}({s_3})={k_{i3}}{s_3}^2+{k_{i9}}{s_3}+{k_{i10}}$. Then we introduce an auxiliary variable ${s_0}$ to convert ${f_i}$ to a homogeneous equation ${F_i}$, , every monomial in $F_i$ is of degree 2. This leads to the following homogeneous equation system: $$\label{equ:31}
\begin{split}
{F_i}  = & {k_{i1}}s_1^2+{k_{i2}}s_2^2+{k_{i4}}{s_1}{s_2}+{p_{i1}}({s_3}){s_0}s_1+ \\
&{p_{i2}}({s_3}){s_0}{s_2}+{p_{i3}}({s_3}){s_0^2}=0, i=1,2,3.
\end{split}$$ If we treat $s_0^2$, $s_1$ and $s_2$ as variables, we can rewrite (\[equ:31\]) as a linear homogeneous system in $s_0^2$, $s_1$ and $s_2$ as follows $$\label{equ:32}
\left[ {\begin{array}{*{20}{c}}
    {{P_{11}}}&{{P_{12}}}&{{P_{13}}}\\
    {{P_{21}}}&{{P_{22}}}&{{P_{23}}}\\
    {{P_{31}}}&{{P_{32}}}&{{P_{33}}}
    \end{array}} \right]\left[ \begin{array}{l}
s_0^2\\
{s_1}\\
{s_2}
\end{array} \right] = {{\bf{0}}_{3 \times 1}},
%{P_{i1}}s_0^{2} + {P_{i2}}s_1 + {P_{i3}}s_2=0, i=1,2,3,$$ where $P_{i1}=p_{i3}(s_3)$, $P_{i2}=k_{i1}{s_1} + k_{i4}{s_2} + p_{i1}(s_3)s_0$, $P_{i3}=k_{i2}s_2 + p_{i2}(s_3)$, $i=1,2,3$. 