Skip to main content
SpringerLink
Log in

Multidimensional Residues and Polynomial Equations

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We describe some recent results of real algebraic geometry which are obtained using multidimensional residues. Our main focus is on the algebraic formulas for topological invariants such as the mapping degree and Euler characteristic. Our proof of the algebraic formula for the mapping degree is based on the properties of a multidimensional logarithmic residue, which are discussed in detail. Several recent applications of the main results are also presented. In particular, we discuss applications to the topological study of quadratic mappings, configuration spaces, quadratic Poisson structures, and Gaussian random polynomials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. A. Agrachev and R. Gamkrelidze, “Quadratic mapping and smooth vector-functions: Euler characteristic of the level set,” In: Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Fundamental Directions [in Russian], 35, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1989), pp. 179–239.

    Google Scholar 

  2. L. Aisenberg and A. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  3. T. Aliashvili, “Topological invariants of random polynomials,” In: Banach Center Publ., 62, 19–28 (2004).

    MATH  MathSciNet  Google Scholar 

  4. T. Aliashvili, “Counting real roots of polynomial endomorphisms,” J. Math. Sci., 118, 5325–5346 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  5. V. Arnold, “Index of a singular point of a vector field, the Petrovsky-Oleynik inequality, and mixed Hodge structures,” Funktsional. Anal. Prilozhen., 12, 1–14 (1978).

    MATH  Google Scholar 

  6. V. Arnold, “Critical points on the boundary and invariants of reflection groups,” Usp. Mat. Nauk, 34, 3–38 (1979).

    MATH  Google Scholar 

  7. V. Arnold, A. Varchenko, and S. Gusein-Zade, Singularities of Differentiable Mappings. Vol. 1 [in Russian], Nauka, Moscow (1982).

    Google Scholar 

  8. M. Berry and J. Hannay, “Umbilic points of Gaussian random surfaces,” J. Phys. A., 10, 1809–1821 (1977).

    Google Scholar 

  9. A. Bharucha-Reid and M. Sambandham, Random Polynomials, Academic Press, New York (1980).

    Google Scholar 

  10. J. Bochnak, M. Coste, and M.-F. Roy, Geometrie Algebrique Reelle, Springer, Berlin (1990).

    Google Scholar 

  11. J. Bruce, “Euler characteristics of real varieties,” Bull. London Math. Soc., 22, 213–219 (1990).

    MathSciNet  Google Scholar 

  12. V. Castellanos Vargas, “The index of non algebraically isolated singularities,” Bol. Soc. Mat. Mexicana, 8, 141–147 (2002).

    MATH  MathSciNet  Google Scholar 

  13. A. Edelman and E. Kostlan, “How many roots of a random polynomial are real?” Bull. Amer. Math. Soc., 32, 1–37 (1995).

    MathSciNet  MATH  Google Scholar 

  14. D. Eisenbud and H. Levine, “An algebraic formula for the degree of a C ∞ map germ,” Ann. Math., 106, 19–44 (1977).

    Article  MathSciNet  Google Scholar 

  15. G.-M. Greuel and H. Hamm, “Invarianten quasihomogener Durchschnitte,” Invent. Math., 49, 67–86 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Griffiths and J. Harris, Principles of Algebraic Geometry, J. Wiley and Sons (1978).

  17. M. Hirsh, Differential Topology, Springer, Berlin (1976).

    Google Scholar 

  18. I. Ibragimov and S. Podkorytov, “On random algebraic surfaces,” Dokl. Ross. Akad. Nauk, 343, 734–736 (1995).

    MathSciNet  Google Scholar 

  19. M. Kac, “On the average number of real roots of a random algebraic equation,” Bull. Amer. Math. Soc., 49, 314–320 (1943).

    MATH  MathSciNet  Google Scholar 

  20. Y. Kamiyama, “The Euler characteristic of the moduli space of polygons in higher-dimensional Euclidean space,” Kyushu J. Math. 54, 333–369 (2000).

    MATH  MathSciNet  Google Scholar 

  21. M. Kapovich and J. Millson, “On the moduli spaces of polygons in the Euclidean plane,” J. Differential Geom., 42, 133–164 (1995).

    MathSciNet  MATH  Google Scholar 

  22. M. Kapovich and J. Millson, “Universality theorems for configuration spaces of planar linkages,” Topology, 41, 1051–1107 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  23. G. Khimshiashvili, “On the local degree of a smooth mapping,” Bull. Acad. Sci. Georgian SSR, 85, 309–312 (1977).

    MATH  Google Scholar 

  24. G. Khimshiashvili, “Euler characteristic of manifold and critical points of smooth functions,” In: Trudy Tbiliss. Mat. Inst. Razmadze, 85, 123–141 (1982).

    MathSciNet  Google Scholar 

  25. G. Khimshiashvili, “On the cardinality of a semi-algebraic subset,” Georgian Math. J., 1, 111–120 (1994).

    Google Scholar 

  26. G. Khimshiashvili, “Signature formulae for topological invariants,” Proc. A. Razmadze Math. Inst., 125, 1–121 (2001).

    MATH  MathSciNet  Google Scholar 

  27. G. Khimshiashvili, “On configuration spaces of planar pentagons,” Nauch. Zap. Semin. Sankt. Peterb. Otd. Mat. Inst. Steklova, 292, 121–128 (2002).

    Google Scholar 

  28. G. Khimshiashvili and R. Przybysz, “On certain super-integrable Hamiltonian systems,” J. Dynam. Control Systems, 8, 217–244 (2002).

    MathSciNet  MATH  Google Scholar 

  29. G. Khimshiashvili and A. Ushveridze, “On the average topological degree of random polynomials,” Bull. Georgian Acad. Sci., 159, 385–388 (1999).

    MathSciNet  MATH  Google Scholar 

  30. A. Khovansky, “The index of a polynomial vector field,” Funktsional. Anal. Prilozhen., 13, 49–58 (1979).

    Google Scholar 

  31. M. Krein and M. Neimark, The Method of Symmetric and Hermitian Forms for the Theory of Separation of Roots of Algebraic Polynomials [in Russian], GNTI, Kharkov (1936).

    Google Scholar 

  32. S. Lang, Algebra, Addison-Wesley, Reading, Mass. (1965).

    MATH  Google Scholar 

  33. A. Lecki and Z. Szafraniec, “An algebraic method for calculating the topological degree,” Banach Center Publ., 35, 73–83 (1996).

    MathSciNet  Google Scholar 

  34. A. McLennan, “The expected number of real roots of a multihomogeneous system of polynomial equations,” Amer. J. Math., 124, 49–73 (2002).

    MATH  MathSciNet  Google Scholar 

  35. N. Nakanishi, “Poisson cohomology of plane quadratic Poison structures,” Publ. Res. Inst. Math. Sci., 33, 73–89 (1997).

    MATH  MathSciNet  Google Scholar 

  36. J. Nash, “Real algebraic manifolds,” Ann. of Math., 56, 405–421 (1952).

    MATH  MathSciNet  Google Scholar 

  37. A. Odessky and V. Rubtsov, “Polynomial Poisson algebras with regular structure of symplectic leaves,” Teor. Mat. Fiz., 133, 3–23 (2002).

    Google Scholar 

  38. V. Palamodov, “On the multiplicity of holomorphic mappings,” Funktsional. Anal. Prilozhen., 1, 54–65 (1967).

    MATH  MathSciNet  Google Scholar 

  39. V. Palamodov, “Remarks on differentiable mappings,” Funktsional. Anal. Prilozhen., 6, 52–61 (1972).

    MATH  MathSciNet  Google Scholar 

  40. S. Podkorytov, “The mean value of the Euler characteristic of a random algebraic hypersurface,” Algebra Analiz, 11, 185–193 (1999).

    MathSciNet  Google Scholar 

  41. M. Postnikov, Stable Polynomials [in Russian], Nauka, Moscow (1982).

    Google Scholar 

  42. R. Przybysz, “On one class of exact Poisson structures,” J. Math. Phys., 42, 1913–1920 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  43. G. Scheja and U. Storch, “Uber Spurfunktionen bei vollstandigen Durchschnitten,” J. Reine Angew. Math., 278/279, 174–190 (1975).

    MathSciNet  Google Scholar 

  44. A. Shiryaev, Probability [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  45. M. Shub and S. Smale, “Complexity of Bezout's theorem II: Volumes and probabilities,” Progr. Math., 109, 267–285 (1993).

    MathSciNet  Google Scholar 

  46. E. Sklyanin, “On some algebraic structures connected with the Yang-Baxter equation,” Funktsional. Anal. Prilozhen., 16, 27–34 (1982).

    MATH  MathSciNet  Google Scholar 

  47. E. Sklyanin, “On some algebraic structures connected with the Yang-Baxter equation: Representations of quantum algebra,” Funktsional. Anal. Prilozhen., 17, 34–48 (1983).

    MATH  MathSciNet  Google Scholar 

  48. E. Spanier, Algebraic Topology, McGraw-Hill, New York (1966).

    MATH  Google Scholar 

  49. Z. Szafraniec, “On the Euler characteristic of analytic and algebraic sets,” Topology, 25, 411–414 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  50. Z. Szafraniec, “The Euler characteristic of algebraic complete intersections,” J. Reine Angew. Math., 397, 194–201 (1989).

    MATH  MathSciNet  Google Scholar 

  51. W. Thurston, “Shapes of polyhedra and triangulations of the sphere,” Geom. Topol. Monogr., 1, 511–549 (1998).

    MATH  MathSciNet  Google Scholar 

  52. V. Trofimov and A. Fomenko, Algebra and Topology of Integrable Hamiltonian Systems [in Russian], Faktorial, Moscow (1995).

    Google Scholar 

  53. A. Tsikh, Multidimensional Residues and Their Applications [in Russian], Nauka, Novosibirsk (1988).

    Google Scholar 

  54. I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhauser, Boston (1994).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A. M. Razmadze Math. Institute, Georgian Academy of Sciences, Georgia

    G. Khimshiashvili

Authors
  1. G. Khimshiashvili
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khimshiashvili, G. Multidimensional Residues and Polynomial Equations. J Math Sci 132, 757–804 (2006). https://doi.org/10.1007/s10958-006-0021-1

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-006-0021-1

Keywords

Search

Navigation