How to solve quadratic equations with three variable?

_{Previously posted at MathOverflow}

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Suppose we have $p$ quadratic equations
begin{eqnarray}
left{
begin{array}{lr}
mathbf{x}^top mathbf{C}_1 mathbf{x} = 1, \
mathbf{x}^top mathbf{C}_2 mathbf{x} = 1, \
quadquad vdots\
mathbf{x}^top mathbf{C}_p mathbf{x} = 1, \
end{array}
right.
label{eq:quadEqus}
end{eqnarray}
where $mathbf{x} in mathbb{R}^3$, $mathbf{C}_1 sim mathbf{C}_p inmathbb{R}^{3 times 3}$ are positive definite symmetric matrices.

Suppose $mathbf{x} = [a,b,c]^top$, and $mathbf{y} = [a^2,b^2,c^2, ab, ac, bc]^top$, it is known that the above quadratic equations can be written as
begin{equation}
mathbf{Ay} = mathbf{1},
label{eq:linearsolver}
end{equation}
where $mathbf{A} in mathbb{R}^{p times 6}$. If $p geq 6$, we can obtain $mathbf{y}$ and further estimate $mathbf{x}$.

Questions:

1. Is there any methods besides $mathbf{Ay}=mathbf{1}$ to solve the above quadratic equations?
2. What is the minimal value of $p$ to guarantee a unique solution of $mathbf{x}$ of the above quadratic system?

Thanks.