Characterization of plane to plane map germs by invariants

We characterize the map germs of corank at most 11 with AA-codimension ≤6≤6 in terms of certain invariants such as multiplicity and the number of cusps of map germs. On the basis of this characterization we present an algorithm to classify the map germs of corank at most 11 from (C2,0)(C2,0) to (C2,0)(C2,0) with AA-codimension ≤6≤6 and also give its implementation in the computer algebra system SINGULAR.

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  • [1] S. Aslam, M.A. Binyamin, and G. Pfister, Recognition of unimodal map germs from the plane to the plane by invariants, Int. J. Algebra Comput. 28 (7), 1199–1208, 2018.
  • [2] M.A. Binyamin, H. Mahmood, and S. Kanwal, On the classification of simple maps from the plane to the plane, J. Algebra Appl. 16 (10), 2017, 1750199.
  • [3] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, SINGULAR 4-1-0 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de, 2017.
  • [4] T. Gaffney, The structure of TAf classification and an application to differential geometry, In singularities, Part I, Proc. Sympos. in Pure Math., Amer. Math. Soc. 40, 409–427, 1983.
  • [5] Y. Kabata, Recognition of plane-to-plane map-germs, Topol. Appl. 202, 216–238, 2016.
  • [6] J.H. Rieger, Families of maps from the plane to the plane, J. London Math. Soc. 26 (2), 351–369, 1987.
  • [7] J.H. Rieger, A-unimodal map-germs into the plane, Hokkaido Math. J. 33, 47–64, 2004.
  • [8] K. Saji, Criteria for singularities of smooth maps from the plane into the plane and their applications, Hiroshima Math. J. 40, 229–239, 2010.