Simon Telen
Abstract
In this poster we present the results of [10]. We consider the problem of finding the common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to compute the structure of the quotient ring R/I from the cokernel of a resultant map. This leads to what we call Truncated Normal Forms (TNFs). Algorithms for generic dense and sparse systems follow from the classical resultant constructions. In the presented framework, the concept of a border basis is generalized by relaxing the conditions on the set of basis elements. This allows for algorithms to adapt the choice of basis in order to enhance the numerical stability. We present such an algorithm. The numerical experiments show that the methods allow to compute all zeros of challenging systems (high degree, with a large number of solutions) in small dimensions with high accuracy.
- E. Cattani, D. A. Cox, G. Chèze, A. Dickenstein, M. Elkadi, I. Z. Emiris, A. Galligo, A. Kehrein, M. Kreuzer, and B. Mourrain. Solving polynomial equations: foundations, algorithms, and applications (Algorithms and Computation in Mathematics). 2005. Google Scholar
Digital Library
- D. A. Cox, J. Little, and D. O'Shea. Ideals, varieties, and algorithms, volume 3. Springer, 1992.Google Scholar
- D. A. Cox, J. Little, and D. O'Shea. Using algebraic geometry, volume 185. Springer Science & Business Media, 2006.Google Scholar
- M. Elkadi and B. Mourrain. Introduction à la résolution des systèmes polynomiaux, volume 59 of Mathématiques et Applications. Springer, 2007.Google Scholar
- I. Z. Emiris and B. Mourrain. Matrices in Elimination Theory. Journal of Symbolic Computation, 28(1--2):3--44, 1999.Google Scholar
- M. Joswig, B. Müller, and A. Paffenholz. polymake and lattice polytopes. In 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), Discrete Math. Theor. Comput. Sci. Proc., AK, pages 491--502. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2009.Google Scholar
- B. Mourrain. A New Criterion for Normal Form Algorithms. In Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, LNCS, pages 430--443, London, UK, 1999. Springer-Verlag. Google ScholarDigital Library
- H. J. Stetter. Matrix eigenproblems are at the heart of polynomial system solving. ACM SIGSAM Bulletin, 30(4):22--25, 1996. Google ScholarDigital Library
- B. Sturmfels. Solving Systems of Polynomial Equations. Number 97 in CBMS Regional Conferences. Amer. Math. Soc., 2002.Google Scholar
- S. Telen, B. Mourrain, and M. Van Barel. Solving polynomial systems via a stabilized representation of quotient algebras. arXiv preprint arXiv:1711.04543, 2017.Google Scholar
- S. Telen and M. Van Barel. A stabilized normal form algorithm for generic systems of polynomial equations. Journal of Computational and Applied Mathematics, 342:119--132, 2018.Google ScholarCross Ref
- J. Verschelde. Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Transactions on Mathematical Software (TOMS), 25(2):251--276, 1999. Google ScholarDigital Library
Index Terms
- Truncated normal forms for solving polynomial systems
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