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Given a bivariate polynomial system with variables $(x, y, z)$ like

(1) $ f_1 = x * a_1 + y * a_2 + z * a_3 = 0 $

(2) $ f_2 = x * a_4 + y * a_5 + z * a_6 = 0$

(3) $ f_3 = x^2 + y^2 - 1 = 0$

how do I determine the solutions/roots numerically?

So far, I found out that I can reformulate the equation system into a matrix and solve it for its eigenvalues, which correspond to the roots. I found some resources on this topic and tried to understand them (like this or this). However, they are really theoretic and I need a lot more time to understand them. Is there maybe a easier hands-on solution someone could point me at? Can maybe someone explain how to construct the matrix to solve for the eigenvalues?

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You want to find a vector in the intersection of two planes that also lies on a cylinder. $\langle \vec a,\vec x\rangle=⟨\vec b,\vec x ⟩=0$ has as solution the line in direction $\vec a\times\vec b$, and if that is not the vertical line, it is easy to find the multiples that intersect the cylinder over the unit circle in the $xy$-plane.

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    $\begingroup$ Alternatively, one could find the equations of ellipses (or straight lines) resulting from intersection of each of the planes with the cylinder; and after that find intersections of those curves. I'd be surprised if this problem cannot be solved analytically. $\endgroup$ Commented Aug 29, 2023 at 19:53

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