Overdetermined systems
A system of polynomial equations $f=(f_1(x_1,\ldots, x_m),\ldots, f_n(x_1,\ldots,x_m))$ is called overdetermined, if it has more equations than variables; i.e., when $n>m$. HomotopyContinuation.jl can solve overdetermined systems. Here is a simple example.
$$f(x,y,z) = \begin{bmatrix} xz-y^2 \\ y-z^2 \\ x-yz \\ x + y + z + 1\end{bmatrix}.$$
This system has 4 equation in 3 variables. One might expect that it has no solution, but actually it has solutions, as is explained here.
The Julia code is as follows
using HomotopyContinuation
@var x y z
solve([x*z - y^2, y - z^2, x - y*z, x + y + z + 1])
Result with 3 solutions
=======================
• 5 paths tracked
• 3 non-singular solutions (1 real)
• 2 excess solutions
• random_seed: 0xf2aeb943
• start_system: :polyhedral
Here, the term excess solutions refers to artificially added solutions in order to make the overdetermined system a square system.