rank and determinant of system of quadratic equations

Question

We have the notion of rank and determinant defined for linear system of equations. is the same defined for quadratic system of equations?

Foe eg: I have system of quadratic equations. the matrix part of this is defined some thing like this as: $a_{00}x_1^{2}+a_{01}x_1x_2+a_{02}x_2^{2}= c1$

$a_{10}x_1^{2}+a_{11}x_1x_2+a_{12}x_2^{2}=c2$

$a_{00}x_1^{2}+a_{01}x_1x_2+a_{02}x_2^{2}=c_3$

in the above can I find the determinant of matrix $A=(a_{ij})\text{ and rank of }A=(a_{ij})$.

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Equivalently, how do we conclude the system of quadratic equation is solvable or not?

Thank you!

Answer 1

You clearly want $x_1^2$ and $x_2^2$ to be non-negative. Otherwise treat the system as a linear one with variables $z_1=x_1^2$, $z_2=x_1x_2$ and $z_3=x_2^2$ and check if $z_2^2=z_1z_3$.