Abstract
Estimating the number of isolated roots of a polynomial system is not only a fundamental study theme in algebraic geometry but also an important subproblem of homotopy methods for solving polynomial systems. For the mixed trigonometric polynomial systems, which are more general than polynomial systems and rather frequently occur in many applications, the classical Bézout number and the multihomogeneous Bézout number are the best known upper bounds on the number of isolated roots. However, for the deficient mixed trigonometric polynomial systems, these two upper bounds are far greater than the actual number of isolated roots. The BKK bound is known as the most accurate upper bound on the number of isolated roots of a polynomial system. However, the extension of the definition of the BKK bound allowing it to treat mixed trigonometric polynomial systems is very difficult due to the existence of sine and cosine functions. In this paper, two new upper bounds on the number of isolated roots of a mixed trigonometric polynomial system are defined and the corresponding efficient algorithms for calculating them are presented. Numerical tests are also given to show the accuracy of these two definitions, and numerically prove they can provide tighter upper bounds on the number of isolated roots of a mixed trigonometric polynomial system than the existing upper bounds, and also the authors compare the computational time for calculating these two upper bounds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Hentenryck P V, McAllester D, and Kapur D, Solving polynomial systems using a branch and prune approach, SIAM J. Numer. Anal., 1997, 34(2): 797–827.
Hong H and Stahl V, Safe starting regions by fixed points and tightening, Computing, 1994, 53(3–4): 323–335.
Morgan A and Sommese A, A homotopy for solving general polynomial systems that respects m-homogeneous structures, Appl. Math. Comput., 1987, 24(2): 101–113.
Morgan A and Sommese A, Computing all solutions to polynomial systems using homotopy continuation, Appl. Math. Comput., 1987, 24(2): 115–138.
Li T Y, Sauer T, and Yorke J A, The random product homotopy and deficient polynomial systems, Numer. Math., 1987, 51(5): 481–500.
Verschelde J and Haegemans A, The GBQ-algorithm for constructing start systems of homotopies for polynomial systems, SIAM J. Numer. Anal., 1993, 30(2): 583–594.
Bernstein D N, The number of roots of a system of equations, Funkcional. Anal. i Priložen., 1975, 9(3): 1–4.
Kušnirenko A G, Newton polyhedra and Bezout’s theorem, Funkcional. Anal. i Priložen., 1976, 10(3): 82–83.
Hovanskiĭ A G, Newton polyhedra, and the genus of complete intersections, Funkcional. Anal. i Priložen., 1978, 12(1): 51–61.
Rojas J M, A convex geometric approach to counting the roots of a polynomial system, Theoret. Comput. Sci., 1994, 133(1): 105–140.
Li T Y and Wang X, The BKK root count in C n, Math. Comp., 1996, 65(216): 1477–1484.
Huber B and Sturmfels B, Bernstein’s theorem in affine space, Discrete Comput. Geom., 1997, 17(2): 137–141.
Gritzmann P and Klee V, On the complexity of some basic problems in computational convexity. II. Volume and mixed volumes, Polytopes: Abstract, Convex and Computational (Scarborough, ON, 1993), Vol. 440 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 1994, 373–466.
Dyer M, Gritzmann P, and Hufnagel A, On the complexity of computing mixed volumes, SIAM J. Comput., 1998, 27(2): 356–400.
Verschelde J, Homotopy continuation methods for solving polynomial systems, PhD thesis, Katholieke Universiteit Leuven (Belgium), 1996.
Li T Y and Li X, Finding mixed cells in the mixed volume computation, Found. Comput. Math., 2001, 1(2): 161–181.
Li T Y, Numerical solution of polynomial systems by homotopy continuation methods, Handb. Numer. Anal., 2003, XI: 209–304.
Gao T, Li T Y, and Wu M, Algorithm 846: MixedVol: A software package for mixed-volume computation, ACM Trans. Math. Software, 2005, 31(4): 555–560.
Sommese A and Wampler C W, The Numerical Solution of Systems of Polynomials, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.
Mizutani T, Takeda A, and Kojima M, Dynamic enumeration of all mixed cells, Discrete Comput. Geom., 2007, 37(3): 351–367.
Mizutani T and Takeda A, DEMiCs: A software package for computing the mixed volume via dynamic enumeration of all mixed cells, Software for Algebraic Geometry, Springer, New York, 2008, 59–79.
Dong B and Yu B, Homotopy method for solving mixed trigonometric polynomial systems, J. Inf. Comput. Sci., 2007, 4(2): 505–513.
Gao T A and Wang Z K, A homotopy algorithm for finding the zeros of a class of trigonometric polynomials, Numer. Math. J. Chinese Univ., 1990, 12(4): 297–303.
Yu B and Dong B, A hybrid polynomial system solving method for mixed trigonometric polynomial systems, SIAM J. Numer. Anal., 2008, 46(3): 1503–1518.
Dong B, Yu B, and Yu Y, A symmetric homotopy and hybrid polynomial system solving method for mixed trigonometric polynomial systems, Mathematics of Computation, 2014, 83: 1847–1868.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported in part by the National Natural Science Foundation of China under Grant Nos. 11101067 and 11571061, Major Research Plan of the National Natural Science Foundation of China under Grant No. 91230103 and the Fundamental Research Funds for the Central Universities.
This paper was recommended for publication by Editor ZHANG Yang.
Rights and permissions
About this article
Cite this article
Jiao, L., Dong, B. & Yu, B. Efficiently counting affine roots of mixed trigonometric polynomial systems. J Syst Sci Complex 30, 967–982 (2017). https://doi.org/10.1007/s11424-017-5214-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-017-5214-9