How to solve a system of polynomial equations

For this guide, we're going to walk through an illustrative example

Requirements

If you have not yet installed HomotopyContinuation.jl, please consider the installation guide.

Solve your first system of equations

Consider the following simple system of two polynomials in two variables.

$$ f=\begin{bmatrix}x^2+2y \\ y^2-2 \end{bmatrix} $$

Solving the equation $f=0$ can be accomplished as follows

using HomotopyContinuation # load the package into the current Julia session
@var x y; # declare the variables x and y
f = System([x^2 + 2y, y^2 - 2]) # construct system f
result = solve(f) # solve f

After the computation has finished, you should see the following output.

Result with 4 solutions
=======================
• 4 paths tracked
• 4 non-singular solutions (2 real)
• random seed: 0x09c7d125
• start_system: :polyhedral

We see that $f$ has two real zeros. They are

julia> realsolutions(result)
2-element Array{Array{Float64,1},1}:
 [1.68179, -1.41421]
 [-1.68179, -1.41421]

It is possible to interrupt the computations using ctrl+c. All solutions that have been computed before the interruption will be returned.

What else should I know?

The next guide explains in greater detail how to use HomotopyContinuation.jl. You should also check the rest of our detailed Guides for learning more about the full power of homotopy continuation. Furthermore, our Reference documentation lists all options of solve(f).