solveSystem(List) -- solve a square system of polynomial equations

Synopsis

Usage:
s = solveSystem F
Function: solveSystem
Inputs:
- a list, F, polynomials with complex coefficients
Optional inputs:
- Software => ..., -- specify internal or external software
Outputs:
- a list, s, all complex solutions to the system F=0

Description

Solve a system of polynomial equations using homotopy continuation methods.

i1 : R = CC[x,y];

i2 : F = {x^2+y^2-1, x*y};

i3 : solveSystem F 

o3 = {{1, 1.04231e-15-4.55746e-16*ii}, {-1, -1.04231e-15+4.55746e-16*ii},
     ------------------------------------------------------------------------
     {6.93878e-16+6.71441e-16*ii, 1}, {-7.03852e-16-6.49215e-16*ii, -1}}

o3 : List

The output contains all Points obtained at the end of homotopy paths when tracking a total-degree homotopy. In particular, this means that solving the system that has fewer than Bezout bound many solutions will produce points that are not marked as regular. See track for detailed examples.

Caveat

The system is assumed to be square (#equations = #variables) and to have finitely many solutions.