I have a system of $N$ equations with $n$ complex variables. Each equation has the form $a x_n^2 + b Re(x_nx_{n-1}) + ...+ c x_{n-1}^2 + d Re(x_{n-1}x_{n-2}) +...+ fx_1^2 = H$, where $H$ is a real number. For the question's sake, let's use $n=2$ and $N=4$. This will give us a system that looks like this: $$ A_{11} x^2 + A_{12} xy\cos\eta + A_{13}y^2 = H_1$$ $$ A_{21} x^2 + A_{22} xy\cos\eta + A_{23}y^2 = H_2$$ $$ A_{31} x^2 + A_{32} xy\cos\eta + A_{33}y^2 = H_3$$ $$ A_{41} x^2 + A_{42} xy\cos\eta + A_{43}y^2 = H_4$$ My idea is that I can express the problem as: $$ \begin{pmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \\ A_{41} & A_{42} & A_{43} \\ \end{pmatrix} \begin{pmatrix} z_1 \\ z_2\\ z_3 \end{pmatrix} = \begin{pmatrix} H_1 \\ H_2\\ H_3 \\ H_4 \end{pmatrix} $$ Which now has the form $A \textbf{z}= H$. This form, I believe, allows me to test for linear independence by solving $A \textbf{z}= 0$. Is this line of thought correct? Furthermore, I would like to add the following two conjectures:
- If the system of equations has infinitely many solutions, then the original quadratic system also has infinitely many solutions.
- If the system of equations has a unique solution, then the original system of equations could have a finite set of solutions
I am not sure how to test these conjectures. In my mind, I am thinking of this problem as a set of parabolic lines that may intersect. The question that I am answering by using the method above is whether the lines are on top of each other. This method doesn't tell me where they intersect.