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Suppose we have a system of $p$ quadratic equations about $\mathbf{x} \in \mathbb{R}^3$ and $\mathbf{x} > 0$

$$ \left\{ \begin{array}{lr} \mathbf{x}^\top \mathbf{C}_1 \mathbf{x} = 1, \\ \mathbf{x}^\top \mathbf{C}_2 \mathbf{x} = 1, \\ \quad\quad \vdots\\ \mathbf{x}^\top \mathbf{C}_p \mathbf{x} = 1, \\ \end{array} \right.$$

where matrices $\mathbf{C}_1, \dotsc, \mathbf{C}_p \in\mathbb{R}^{3 \times 3}$ are symmetric and positive definite.

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?
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    $\begingroup$ What do you mean by "What is the minimal value of $p$ to guarantee a unique solution"? Do you mean the minimal value for which a solution can be unique? No matter how big $p$ is, $\mathbf A$ could be, say, the $0$ matrix, in which case there are no solutions, or some low-rank matrix, in which case there could be multiple solutions. $\endgroup$
    – LSpice
    Jul 21 '20 at 13:23
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    $\begingroup$ If $\mathbf x$ is a solution then so is $-\mathbf x$, so you can't ever have uniqueness. Up to sign you can certainly recover $\mathbf x$ from $\mathbf y$ so the condition for having only two solutions is indeed just that the matrix has full rank. $\endgroup$
    – lambda
    Jul 21 '20 at 13:42
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    $\begingroup$ Have you tried in $\mathbb{R}^2$ first? $\endgroup$ Jul 21 '20 at 15:48
  • $\begingroup$ Related: mathoverflow.net/q/308163 $\endgroup$ Jul 21 '20 at 16:06
  • $\begingroup$ @LSpice Let's say $\mathbf{A}$ is always full rank in this case. I want to know how many quadratic equations are required to obtain $\mathbf{x} \in \mathbb{R}^3$, is p=3 enough? $\endgroup$
    – heng
    Jul 22 '20 at 0:31

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