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Discrete & Computational Geometry

, Volume 62, Issue 4, pp 788–812 | Cite as

On the Multiplicity of Isolated Roots of Sparse Polynomial Systems

  • María Isabel Herrero
  • Gabriela JeronimoEmail author
  • Juan Sabia
Article
  • 37 Downloads

Abstract

We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First, we consider sets of supports such that the origin is an isolated root of the corresponding generic system and prove formulas for its multiplicity. Then, we apply these formulas to solve the problem in the general case, by showing that the multiplicity of an arbitrary affine isolated zero of a generic system with given supports equals the multiplicity of the origin as a common zero of a generic system with an associated family of supports. The formulas obtained are in the spirit of the classical Bernstein’s theorem, in the sense that they depend on the combinatorial structure of the system, namely, geometric numerical invariants associated to the supports, such as mixed volumes of convex sets and, alternatively, mixed integrals of convex functions.

Keywords

Sparse polynomial systems Multiplicity of zeros Newton polytopes Mixed volumes and mixed integrals 

Mathematics Subject Classification

13H15 14Q99 14C17 

Notes

Acknowledgements

The authors wish to thank the referees for their careful reading and helpful suggestions.

References

  1. 1.
    Bernstein, D.N.: The number of roots of a system of equations. Funct. Anal. Appl. 9(3), 183–185 (1975)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bürgisser, P., Scheiblechner, P.: On the complexity of counting components of algebraic varieties. J. Symb. Comput. 44(9), 1114–1136 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cattani, E., Dickenstein, A.: Counting solutions to binomial complete intersections. J. Complexity 23(1), 82–107 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cox, D.A., Little, J., O’Shea, D.: Using Algebraic Geometry. Graduate Texts in Mathematics, vol. 185. New York: Springer (1998)Google Scholar
  5. 5.
    Cueto, M.A., Dickenstein, A.: Some results on inhomogeneous discriminants. In: Ferrer Santos, W. et al. (eds.) Proceedings of the XVIth Latin American Algebra Colloquium. Biblioteca de la Revista Matemática Iberoamericana, pp. 41–62. Revista Matemática Iberoamericana, Madrid (2007)Google Scholar
  6. 6.
    Dayton, B.H., Zeng, Z.: Computing the multiplicity structure in solving polynomial systems. In: Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation (ISSAC’05), pp. 116–123. ACM, New York (2005)Google Scholar
  7. 7.
    Emiris, I.Z., Verschelde, J.: How to count efficiently all affine roots of a polynomial system. Discrete Appl. Math. 93(1), 21–32 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Birkhäuser, Boston (1994)Google Scholar
  9. 9.
    Herrero, M.I., Jeronimo, G., Sabia, J.: Computing isolated roots of sparse polynomial systems in affine space. Theor. Comput. Sci. 411(44–46), 3894–3904 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Herrero, M.I., Jeronimo, G., Sabia, J.: Affine solution sets of sparse polynomial systems. J. Symb. Comput. 51, 34–54 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Huber, B., Sturmfels, B.: Bernstein’s theorem in affine space. Discrete Comput. Geom. 17(2), 137–141 (1997)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kaveh, K., Khovanskii, A.G.: Convex bodies and multiplicities of ideals. Proc. Steklov Inst. Math. 286(1), 268–284 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Khovanskii, A.G.: Newton polyhedra and toroidal varieties. Funct. Anal. Appl. 11(4), 289–296 (1978)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32(1), 1–31 (1976)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kushnirenko, A.G.: Newton polytopes and the Bézout theorem. Funct. Anal. Appl. 10(3), 233–235 (1976)CrossRefGoogle Scholar
  16. 16.
    Li, T.Y., Wang, X.: The BKK root count in \(\mathbb{C}^n\). Math. Comput. 65(216), 1477–1484 (1996)Google Scholar
  17. 17.
    Macaulay, F.S.: The Algebraic Theory of Modular Systems. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1916)zbMATHGoogle Scholar
  18. 18.
    Mondal, P.: Intersection multiplicity, Milnor number and Bernstein’s theorem (2016). arXiv:1607.04860
  19. 19.
    Oka, M.: Non-Degenerate Complete Intersection Singularity. Actualités Mathématiques. Hermann, Paris (1997)zbMATHGoogle Scholar
  20. 20.
    Philippon, P., Sombra, M.: Hauteur normalisée des variétés toriques projectives. J. Inst. Math. Jussieu 7(2), 327–373 (2008)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Philippon, P., Sombra, M.: A refinement of the Bernštein–Kušnirenko estimate. Adv. Math. 218(5), 1370–1418 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rojas, J.M.: A convex geometric approach to counting the roots of a polynomial system. Theor. Comput. Sci. 133(1), 105–140 (1994)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rojas, J.M., Wang, X.: Counting affine roots of polynomial systems via pointed Newton polytopes. J. Complexity 12(2), 116–133 (1996)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Stetter, H.J.: Numerical Polynomial Algebra. SIAM, Philadelphia (2004)CrossRefGoogle Scholar
  25. 25.
    Teissier, B.: Monômes, volumes et multiplicités. In: Tráng, L.D. (ed.) Introduction à laThéorie des Singularités, II. Travaux en Cours, vol. 37, pp. 127–141. Hermann, Paris (1988)Google Scholar

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Authors and Affiliations

  1. 1.Departamento de Matemática, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Instituto de Investigaciones Matemáticas Luis A. Santaló (IMAS)CONICET-Universidad de Buenos AiresBuenos AiresArgentina
  3. 3.Ciclo Básico Común, Departamento de Ciencias ExactasUniversidad de Buenos AiresBuenos AiresArgentina

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