RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor
Subscription
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Uspekhi Mat. Nauk:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Uspekhi Mat. Nauk, 1974, Volume 29, Issue 2(176), Pages 11–49 (Mi umn4352)  

This article is cited in 52 scientific papers (total in 53 papers)

Normal forms of functions in neighbourhoods of degenerate critical points

V. I. Arnol'd


Abstract: An analysis of the normal forms to which functions can be reduced in neighbourhoods of degenerate critical points shows that many of them are quasihomogeneous or semiquasihomogeneous. A semiquasihomogeneous function is a sum of a quasihomogeneous (or weighted homogeneous) polynomial with an isolated critical point and summands of a higher degree of quasihomogeneity. The normal form to which a semiquasihomogeneous function can be reduced is described in terms of the local ring of the gradient mapping given by the quasihomogeneous part of the function. The number of parameters in this normal form is called the inner modality of the quasihomogeneous part. A classification is given of all quasihomogeneous critical points of inner modality 1: up to stable equivalence they are exhausted by three one-parameter families of parabolic singularities and 14 exceptional polynomials, 8 of which are functions of two variables, and 6 functions of three variables.

Full text: PDF file (2240 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 1974, 29:2, 10–50

Bibliographic databases:

Document Type: Article
UDC: 517.5
MSC: 58K05, 58K50, 57R70
Received: 28.09.1973

Citation: V. I. Arnol'd, “Normal forms of functions in neighbourhoods of degenerate critical points”, Uspekhi Mat. Nauk, 29:2(176) (1974), 11–49; Russian Math. Surveys, 29:2 (1974), 10–50

Citation in format AMSBIB
\Bibitem{Arn74}
\by V.~I.~Arnol'd
\paper Normal forms of functions in neighbourhoods of degenerate critical points
\jour Uspekhi Mat. Nauk
\yr 1974
\vol 29
\issue 2(176)
\pages 11--49
\mathnet{http://mi.mathnet.ru/umn4352}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=516034}
\zmath{https://zbmath.org/?q=an:0298.57022|0304.57018}
\transl
\jour Russian Math. Surveys
\yr 1974
\vol 29
\issue 2
\pages 10--50
\crossref{https://doi.org/10.1070/RM1974v029n02ABEH003846}


Linking options:
  • http://mi.mathnet.ru/eng/umn4352
  • http://mi.mathnet.ru/eng/umn/v29/i2/p11

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. I. Arnol'd, “Critical points of smooth functions and their normal forms”, Russian Math. Surveys, 30:5 (1975), 1–75  mathnet  crossref  mathscinet  zmath
    2. V. I. Arnol'd, “Classification of bimodal critical points of functions”, Funct. Anal. Appl., 9:1 (1975), 43–44  mathnet  crossref  mathscinet  zmath
    3. A. G. Kushnirenko, “Newton polyhedron and Milnor numbers”, Funct. Anal. Appl., 9:1 (1975), 71–72  mathnet  crossref  mathscinet  zmath
    4. A. M. Gabrièlov, A. G. Kushnirenko, “Description of deformations with constant milnor number for homogeneous functions”, Funct. Anal. Appl., 9:4 (1975), 329–331  mathnet  crossref  mathscinet  zmath
    5. O. V. Lyashko, “Decomposition of simple singularities of functions”, Funct. Anal. Appl., 10:2 (1976), 122–128  mathnet  crossref  mathscinet  zmath
    6. M V Berry, J Phys A Math Gen, 10:12 (1977), 2061  crossref  mathscinet  zmath  adsnasa
    7. P. Orlik, P. Wagreich, “Algebraic surfaces withk *-action”, Acta Math, 138:1 (1977), 43  crossref  mathscinet  zmath
    8. G. R. Belitskii, “Normal forms relative to the action of a group on a space”, Math. USSR-Izv., 11:5 (1977), 1001–1010  mathnet  crossref  mathscinet  zmath
    9. A. G. Kushnirenko, “Kriterii suschestvovaniya nevyrozhdennoi kvaziodnorodnoi funktsii s zadannymi vesami”, UMN, 32:3(195) (1977), 169–170  mathnet  mathscinet  zmath
    10. V. A. Vassiliev, “Asymptotic exponential integrals, Newton's diagram, and the classification of minimal points”, Funct. Anal. Appl., 11:3 (1977), 163–172  mathnet  crossref  mathscinet  zmath
    11. V. de Alfaro, M. Rasetti, “Structural Stability Theory and Phase Transitions Models”, Fortschr Phys, 26:3 (1978), 143  crossref
    12. R Raghavan, J Phys A Math Gen, 11:4 (1978), 731  crossref  mathscinet  adsnasa
    13. V. I. Danilov, “The geometry of toric varieties”, Russian Math. Surveys, 33:2 (1978), 97–154  mathnet  crossref  mathscinet  zmath
    14. V. I. Arnol'd, “Critical points of functions on a manifold with boundary, the simple Lie groups $B_k$, $C_k$, and $F_4$ and singularities of evolutes”, Russian Math. Surveys, 33:5 (1978), 99–116  mathnet  crossref  mathscinet  zmath
    15. V. I. Arnol'd, “Index of a singular point of a vector field, the Petrovskii–Oleinik inequality, and mixed hodge structures”, Funct. Anal. Appl., 12:1 (1978), 1–12  mathnet  crossref  mathscinet  zmath
    16. I. G. Shcherbak, “Algebras of automorphic forms with three generators”, Funct. Anal. Appl., 12:2 (1978), 156–158  mathnet  crossref  zmath
    17. Etsuo Yoshinaga, Masahiko Suzuki, “Normal forms of non-degenerate quasihomogeneous functions with inner modality≦4”, Invent math, 55:2 (1979), 185  crossref  mathscinet  zmath  isi
    18. V. A. Vassiliev, “Asymptotic behavior of exponential integrals in the complex domain”, Funct. Anal. Appl., 13:4 (1979), 239–247  mathnet  crossref  mathscinet  zmath
    19. J. W. Bruce, “A stratification of the space of cubic surfaces”, Math Proc Camb Phil Soc, 87:3 (1980), 427  crossref  mathscinet  zmath  isi
    20. V. I. Matov, “Unimodal germs of functions on a manifold with a boundary”, Funct. Anal. Appl., 14:1 (1980), 55–57  mathnet  crossref  mathscinet  zmath
    21. K Keller, J Phys A Math Gen, 14:7 (1981), 1719  crossref  mathscinet  zmath  adsnasa  isi
    22. G. J. Griffith, “The “Coxeter curves”, x2/p+y2/p+z2/p=0 for prime values of p”, JGeom, 17:1 (1981), 24  crossref  mathscinet  zmath
    23. J. W. Bruce, “On complex projective hypersurfaces”, Proc Edin Math Soc, 24:2 (1981), 91  crossref  mathscinet  zmath  isi
    24. A. N. Varchenko, “Asymptotic Hodge structure in the vanishing cohomology”, Math. USSR-Izv., 18:3 (1982), 469–512  mathnet  crossref  mathscinet  zmath
    25. Ian Stewart, “Applications of catastrophe theory to the physical sciences”, Physica D: Nonlinear Phenomena, 2:2 (1981), 245  crossref
    26. Jim Callahan, “A geometric model of anorexia and its treatment”, Syst Res, 27:2 (1982), 140  crossref  mathscinet  isi
    27. I Stewart, Rep Prog Phys, 45:2 (1982), 185  crossref  mathscinet  adsnasa  isi
    28. M. G. Soares, P. J. Giblin, “Recognizing singularities of surfaces in P3”, Math Proc Camb Phil Soc, 91:1 (1982), 17  crossref  mathscinet  zmath  isi
    29. A. G. Aleksandrov, “Normal forms of one-dimensional quasihomogeneous complete intersections”, Math. USSR-Sb., 45:1 (1983), 1–30  mathnet  crossref  mathscinet  zmath
    30. J H Hannay, J Phys A Math Gen, 16:2 (1983), L61  crossref  mathscinet  zmath  adsnasa  isi
    31. Yu. N. Zavarovskii, “Normal form of the key function of the generalized Kirchhoff equation”, Russian Math. Surveys, 38:3 (1983), 159–160  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    32. O. V. Lyashko, “Classification of critical points of functions on a manifold with singular boundary”, Funct. Anal. Appl., 17:3 (1983), 187–193  mathnet  crossref  mathscinet  zmath  isi
    33. V. I. Arnol'd, “Vanishing inflections”, Funct. Anal. Appl., 18:2 (1984), 128–130  mathnet  crossref  mathscinet  zmath  isi
    34. Masahiko Suzuki, “The stratum with constant milnor number of a mini-transversal family of a quasihomogeneous function of corank two”, Topology, 23:1 (1984), 101  crossref
    35. Richard W. Ziolkowski, Georges A. Deschamps, “Asymptotic evaluation of high-frequency fields near a caustic: An introduction to Maslov's method”, Radio Sci, 19:4 (1984), 1001  crossref
    36. C.T.C. Wall, “Deformations of real singularities”, Topology, 29:4 (1990), 441  crossref
    37. D. V. Anosov, A. A. Bolibrukh, V. A. Vassiliev, A. M. Vershik, A. A. Gonchar, M. L. Gromov, S. M. Gusein-Zade, V. M. Zakalyukin, Yu. S. Ilyashenko, V. V. Kozlov, M. L. Kontsevich, Yu. I. Manin, A. I. Neishtadt, S. P. Novikov, Yu. S. Osipov, M. B. Sevryuk, Ya. G. Sinai, A. N. Tyurin, L. D. Faddeev, B. A. Khesin, A. G. Khovanskii, “Vladimir Igorevich Arnol'd (on his 60th birthday)”, Russian Math. Surveys, 52:5 (1997), 1117–1139  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    38. V Vassiliev, “On decision trees for orthants”, Information Processing Letters, 62:5 (1997), 265  crossref  elib
    39. V. A. Vassiliev, “On a Problem by M. Kazarian”, Funct. Anal. Appl., 33:3 (1999), 220–221  mathnet  crossref  crossref  mathscinet  zmath  isi
    40. Arnol'd, VI, “Mathematics and physics: mother and daughter or sisters?”, Uspekhi Fizicheskikh Nauk, 169:12 (1999), 1311  mathnet  crossref  isi
    41. Armengol Gasull, Joan Torregrosa, “Small-Amplitude Limit Cycles in Liénard Systems via Multiplicity”, Journal of Differential Equations, 159:1 (1999), 186  crossref
    42. A. D. Bruno, “Self-similar solutions and power geometry”, Russian Math. Surveys, 55:1 (2000), 1–42  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    43. V. V. Goryunov, “Simple Functions on Space Curves”, Funct. Anal. Appl., 34:2 (2000), 129–132  mathnet  crossref  crossref  mathscinet  zmath  isi
    44. E. Brieskorn, A. M. Pratusevich, F. Rothenhäusler, “The combinatorial geometry of singularities and Arnold's series $E$$Z$$Q$”, Mosc. Math. J., 3:2 (2003), 273–333  mathnet  mathscinet  zmath  elib
    45. Jiro Sekiguchi, “Three Dimensional Saito Free Divisors and Singular Curves”, Zhurn. SFU. Ser. Matem. i fiz., 1:1 (2008), 33–41  mathnet  elib
    46. B Konopelchenko, L Martínez Alonso, E Medina, “Hodograph solutions of the dispersionless coupled KdV hierarchies, critical points and the Euler–Poisson–Darboux equation”, J Phys A Math Theor, 43:43 (2010), 434020  crossref
    47. David Chillingworth, Carmen Romero-Fuster, “Remarks on the singularity indices of Arnol'd and Berry”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 94:3-4 (2011), 339  crossref
    48. Antonella Grassi, James Halverson, J.L.. Shaneson, “Matter from geometry without resolution”, J. High Energ. Phys, 2013:10 (2013)  crossref
    49. F.K.anwal Janjua, Gerhard Pfister, “A classifier for simple space curve singularities”, Studia Scientiarum Mathematicarum Hungarica, 51:1 (2014), 92  crossref
    50. M.S.. Marais, Andreas Steenpaß, “The classification of real singularities using Singular Part I: Splitting lemma and simple singularities”, Journal of Symbolic Computation, 2014  crossref
    51. Antonella Grassi, James Halverson, J.L.. Shaneson, “Non-Abelian Gauge Symmetry and the Higgs Mechanism in F-Theory”, Commun. Math. Phys, 2015  crossref
    52. Deeba Afzal, Gerhard Pfister, “A classifier for simple isolated complete intersection singularities”, Studia Scientiarum Mathematicarum Hungarica, 52:1 (2015), 1  crossref
    53. M.S.. Marais, Andreas Steenpaß, “The classification of real singularities using Singular Part II: The structure of the equivalence classes of the unimodal singularities”, Journal of Symbolic Computation, 2015  crossref
  • Успехи математических наук Russian Mathematical Surveys
    Number of views:
    This page:1014
    Full text:396
    References:50
    First page:6

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019