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Symbolic Numerical Polynomial Solving
Solving polynomial systems using Gröbner bases and
triangular sets:
Input: 
Zerodimensional system
f_{1} ,...,
f_{k} in
K[x_{1} ,..., x_{n}]

Output: 
Complex roots of
f_{1} = ... = f_{k} = 0

The algorithm proceeds in 3 steps:
Step 1: 
Compute a reduced lexicographical Gröbner basis of the
ideal I.

Step 2: 
Compute a triangular
system T_{1} ,...,
T_{s} .
V(I) is the union
of the V(T_{i} ).

Step 3: 
Use a numerical solver (e.g. Laguerre's method) to find all zeros of
T_{i} , i=1,..., s .

