Solve nonlinear equations: variables with degree six and degree eight.

Question

Suppose I have two nonlinear equations with two variables $\ell$ and $m$; the variables $\ell$ and $m$ are of degree eight in the first equation and of degree six in the second equation.

How it possible to solve these two nonlinear equations and find the possible values of the variables $\ell$ and $m$?

How many possible solutions for both variables $\ell$ and $m$?

Is there a Matlab code to solve these equations and find the all possible solutions?.

Answer 1

You can try Two Variable Bisection.

$\quad\mathbf2$D Bisection

Bisection in $2$ dimensions amount to finding zeros of two equations in two unknowns, say $F(x,y)=0$ and $G(x,y)=0$ on a rectangular region in the plane. The algorithm is effectively a quadtree decomposition of the region into subregions, each subregion (presumably) having $F\neq0$ or $G\neq0$. I say presumably since the algorithm check the signs of $F$ and $G$ at the vertices of the subregions. If $F$ has the same sign at the four vertices, then the algorithm stops processing that subregion (maybe missing roots).

^{related post:} https://stackoverflow.com/q/3513660/380384