Journal of the Mathematical Society of Japan

On polynomial mappings from the plane to the plane


Full-text: Open access


Let $f:{\mathbb R}^2\longrightarrow {\mathbb R}^2$ be a generic polynomial mapping. There are constructed quadratic forms whose signatures determine the number of positive and negative cusps of $f$.

Article information

J. Math. Soc. Japan, Volume 66, Number 3 (2014), 805-818.

First available in Project Euclid: 24 July 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14P99: None of the above, but in this section
Secondary: 58K05: Critical points of functions and mappings

singularities cusps


KRZYŻANOWSKA, Iwona; SZAFRANIEC, Zbigniew. On polynomial mappings from the plane to the plane. J. Math. Soc. Japan 66 (2014), no. 3, 805--818. doi:10.2969/jmsj/06630805.

Export citation


  • E. Becker and T. Wöermann, On the trace formula for quadratic forms, Contemp. Math., 155, In: Recent Advances in Real Algebraic Geometry and Quadratic Forms, California, 1990/1991, (eds. W. B. Jacob, T. Y. Lam, and R. O. Robson), Amer. Math. Soc., Providence, RI, 1994, pp.,271–291.
  • N. Dutertre and T. Fukui, On the topology of stable maps, J. Math. Soc. Japan, 66 (2014), 161–203.
  • T. Fukuda and G. Ishikawa, On the number of cusps of stable perturbations of a plane-to-plane singularity, Tokyo J. Math., 10 (1987), 375–384.
  • T. Gaffney and D. M. Q. Mond, Cusps and double folds of germs of analytic maps ${\mathbb C}^2\rightarrow {\mathbb C}^2$, J. London Math. Soc. (2), 43 (1991), 185–192.
  • M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Grad. Texts in Math., 14, Springer-Verlag, New York, 1973.
  • G.-M. Greuel, G. Pfister and H. Schönemann, Singular 3.0.2. A Computer Algebra System for Polynomial Computations.
  • H. I. Levine, Mappings of manifolds into the plane, Amer. J. Math., 88 (1966), 357–365.
  • J. A. Moya-Pérez and J. J. Nuño-Ballesteros, The link of a finitely determined map germ from $\bm{R}^2$ to $\bm{R}^2$, J. Math. Soc. Japan, 62 (2010), 1069–1092.
  • P. Pedersen, M.-F. Roy and A. Szpirglas, Counting real zeros in the multivariate case, In: Computational Algebraic Geometry, Nice, 1992, (eds. F. Eyssette and A. Galligo), Progr. Math., 109, Birkhäuser, 1993, pp.,203–224.
  • J. H. Rieger, Families of maps from the plane to the plane, J. London Math. Soc. (2), 36 (1987), 351–369.
  • J. R. Quine, A global theorem for singularities of maps between oriented 2-manifolds, Trans. Amer. Math. Soc., 236 (1978), 307–314.
  • O. Saeki, Topology of Singular Fibers of Differentiable Maps, Lecture Notes in Math., 1854, Springer-Verlag, 2004, Berlin.
  • R. Thom, Les singularités des applications différentiables, Ann. Inst. Fourier, Grenoble, 6 (1955–1956), 43–87.
  • H. Whitney, On singularities of mapping of Euclidean spaces. I. Mappings of the plane into the plane, Ann. of Math. (2), 62 (1955), 374–410.