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Symbolic Numerical Polynomial Solving


Solving polynomial systems using Gröbner bases and triangular sets:
Input: Zero-dimensional system f1 ,..., fk in K[x1 ,..., xn]
Output: Complex roots of f1 = ... = fk = 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 T1 ,..., Ts .
V(I) is the union of the V(Ti ).
Step 3: Use a numerical solver (e.g. Laguerre's method) to find all zeros of Ti , i=1,..., s .

Timings

Paris 7-2-01 http://www.singular.uni-kl.de