# Symbolic-Numeric Solution of Ill-Conditioned Polynomial Systems (Survey Talk Overview) (Invited Talk)

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## Abstract

This is a survey talk about some recent symbolic-numeric techniques to solve ill-conditioned multivariate polynomial systems. In particular, we will concentrate on systems that are over-constrained or have roots with multiplicities, and are given with inexact coefficients. First I give some theoretical background on polynomial systems with inexact coefficients, ill-posed and ill-conditioned problems, and on the objectives when trying to solve these systems. Next, I will describe a family of iterative techniques which, for a given inexact system of polynomials and given root structure, computes the nearest system which has roots with the given structure. Finally, I present a global method to solve multivariate polynomial systems which are near root multiplicities and thus have clusters of roots. The method computes a new system which is “square-free”, i.e. it has exactly one root in each cluster near the arithmetic mean of the cluster. This method is global in the sense that it works simultaneously for all clusters.

The results presented in the talk are joint work with Itnuit Janovitz-Freireich, Bernard Mourrain, Scott Pope, Lajos Rónyai, Olivier Ruatta, and Mark Sciabica.

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## References

- 1.S. E. Hutton, Exact Sums-of-Squares Certificates in Numeric Algebraic Geometry, PhD thesis, North Carolina State University (2011)Google Scholar
- 2.Janovitz-Freireich, I., Rónyai, L., Szántó, Á.: Approximate radical of ideals with clusters of roots. In: ISSAC 2006, pp. 146–153. ACM, New York (2006)Google Scholar
- 3.Janovitz-Freireich, I., Rónyai, L., Szántó, Á.: Approximate radical for clusters: a global approach using Gaussian elimination or SVD. Math. Comput. Sci. 1, 393–425 (2007)Google Scholar
- 4.Janovitz-Freireich, I., Szántó, Á., Mourrain, B., Rónyai, L.: Moment matrices, trace matrices and the radical of ideals. In: ISSAC 2008, pp. 125–132. ACM, New York (2008)Google Scholar
- 5.Janovitz-Freireich, I., Szántó, Á., Mourrain, B., Rónyai, L.: On the Computation of Matrices of Traces and Radicals of Ideals. Submitted to Journal of Symbolic Computation (2009); arXiv:0901.2778Google Scholar
- 6.Karmarkar, N.K., Lakshman, Y.N.: On approximate GCDs of univariate polynomials. Journal of Symbolic Computation 26, 653–666 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
- 7.Mourrain, B.: Isolated points, duality and residues, J. Pure Appl. Algebra 117/118, 469–493 (1997); Algorithms for algebra, Eindhoven (1996)Google Scholar
- 8.Pope, S., Szanto, A.: Nearest multivariate system with given root multiplicities. Journal of Symbolic Computation, 606–625 (2009)Google Scholar
- 9.Ruatta, O.: Dualité algébrique, structures et applications, PhD thesis, Université de la Méditérranée (2002)Google Scholar
- 10.Ruatta, O., Sciabica, M., Szanto, A.: Over-constrained Weierstrass iteration and the nearest consistent system (2009) (accepted for publication) Google Scholar
- 11.Zeng, Z.: Computing multiple roots of inexact polynomials. Mathematics of Computation 74, 869–903 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Zhi, L., Wu, W.: Nearest singular polynomials, J. Symbolic Comput. 26, 667–675 (1998); Symbolic numeric algebra for polynomialsGoogle Scholar