# Solve bivariate polynomial system

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?

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.

• 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. Commented Aug 29, 2023 at 19:53