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

This article is cited in 53 scientific papers (total in 54 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.

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

Bibliographic databases:

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
\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
\jour Russian Math. Surveys
\yr 1974
\vol 29
\issue 2
\pages 10--50

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    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
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    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
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    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
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    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  crossref  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
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