Using monodromy to decompose solution sets of polynomial systems into irreducible components. (English) Zbl 0990.65051

Sommese, A. J.; Verschelde, J.; Wampler, C. W.

Ciliberto, Ciro (ed.) et al., Applications of algebraic geometry to coding theory, physics and computation. Proceedings of the NATO advanced research workshop, Eilat, Israel, February 25-March 1, 2001. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 36, 297-315 (2001).

Summary: To decompose solution sets of polynomial systems into irreducible components, homotopy continuation methods generate the action of a natural monodromy group which partially classifies generic points onto their respective irreducible components. As illustrated by the performance on several text examples, this new method achieves a great increase in speed and accuracy, as well as improved numerical conditioning of the multivariate interpolation problem.
For the entire collection see [Zbl 0971.00013].

MSC:

65H10	Numerical computation of solutions to systems of equations
13P05	Polynomials, factorization in commutative rings
14Q99	Computational aspects in algebraic geometry
68W30	Symbolic computation and algebraic computation

Keywords:

numerical examples; components of solutions; embedding; generic points; homotopy continuation; irreducible components; monodromy group; numerical algebraic geometry; polynomial system; primary decomposition; numerical conditioning; multivariate interpolation