Overdetermined systems

We're tracking a solution of an overdetermined system

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

julia> using HomotopyContinuation
julia> @polyvar x y z
julia> solve([x*z - y^2, y - z^2, x - y*z, x + y + z + 1])
Result with 3 solutions
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• 3 non-singular solutions (1 real)
• 0 singular solutions (0 real)
• 8 paths tracked
• random seed: 157717