Polynomial automorphisms of the affine plane and singularities of algebraic curves

  1. Evelia R. García Barroso 1
  2. Arkadiusz Ploski
  1. 1 Universidad de La Laguna
    info

    Universidad de La Laguna

    San Cristobal de La Laguna, España

    ROR https://ror.org/01r9z8p25

Revista:
Revista de la Academia Canaria de Ciencias. Mathematics section

ISSN: 1130-4723

Año de publicación: 2018

Volumen: XXX

Páginas: 31-54

Tipo: Artículo

GOOGLE SCHOLAR

Otras publicaciones en: Revista de la Academia Canaria de Ciencias. Mathematics section

Resumen

Abhyankar and Moh achieved a major breakthrough in the globalgeometry of the affine plane with their papers [2] and [ 3]. The aim of thisexpository article is to provide an introduction to the Abhyankar-Moh theory.We base our approach on the local theory of algebraic plane curves explained inour previous paper [18], where we reproved the basic properties of approximateroots without resorting to Puiseux series. We pass then to the projective closureof the affine plane in order to prove the Embedding Line Theorem [3] and related results such as the Moh-Ephraim Pencil Theorem and the Abhyankar-Moh Semigroup Theorem.

Referencias bibliográficas

  • [1] Abhyankar, S.S., Expansion techniques in Algebraic Geometry, Tata Institute of Funda- mental Research Lectures on Mathematics and Physics, 57, Tata Institute of Fundamental Research, Bombay, iv+168 pp., 1977.
  • [2] Abhyankar, S.S. and Moh, T.T., Newton-Puiseux expansion and generalized Tschirnhausen transformation I, II, J. reine angew. Math., 260, 47–83; ibid. 261, 29–54, 1973.
  • [3] Abhyankar, S.S. and Moh, T.T., Embeddings of the line in the plane, J. reine angew. Math., 276, 148–166, 1975.
  • [4] Abhyankar, S.S. and Sathaye, A., Uniqueness of plane embeddings of special curves, Proc. Amer. Math. Soc., 124, no. 4, 1061–1069, 1996.
  • [5] Angerm ̈uller, G., Die Wertehalbgruppe einer ebenen irreduziblen algebroiden Kurve, Math. Z., 153, no. 3, 267–282, 1977.
  • [6] Assi, A. and Garc ́ıa-S ́anchez, P.A., Algorithms for curves with one place at infinity, Journal of Symbolic Computation, 74, 475–492, 2016.
  • [7] Barrolleta, R.D., Garc ́ıa Barroso, E.R. and P 4loski, A., On the Abhyankar-Moh inequality, Universitatis Iagellonicae Acta Mathematica, LII, 7–14, 2015.
  • [8] Bresinsky, H., Semigroups corresponding to algebroid branches in the plane, Proc. Amer. Math. Soc., 32, no. 2, 381–384, 1972.
  • [9] Campillo, A., Algebroid curves in positive characteristic, Lecture Notes in Mathematics, 813. Springer Verlag, Berlin, v+168 pp., 1980.
  • [10] Campillo, A. and Farr ́an, J.I., Computing Weierstrass Semigroups and the Feng-Rao Dis- tance from Singular Plane Models, Finite Fields and Their Applications, 6, 71–92, 2000.
  • [11] Casas-Alvero, E., Singularities of plane curves, London Mathematical Society Lecture Note Series, 276, Cambridge University Press, Cambridge, xvi+345 pp., 2000.
  • [12] Chang, H. C. and Wang, L. C., An intersection-theoretical proof of the embedding line theorem, J. Algebra, 161, no. 2, 467–479, 1993.
  • [13] Ephraim, R., Special polars and curves with one place at infinity, Singularities, Part 1 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 353–359, 1983.
  • [14] Fujimoto, M. and Suzuki, M., Construction of affine plane curves with one place at infinity, Osaka J. Math., 39 no. 4, 1005–1027, 2002.
  • [15] Fujimoto, M., Suzuki, M. and Yokoyama, K., On polynomial curves in the affine plane, Osaka J. Math., 43, no. 3, 597–608, 2006
  • [16] Ganong, R., On plane curves with one place at infinity, J. reine angew. Math., 307/308, 173–193, 1979.
  • [17] Garc ́ıa Barroso, E.R., Gwo ́ zdziewicz, J. and P4loski, A., Semigroups corresponding to branches at infinity of coordinate lines in the affine plane, Semigroup Forum, 92, 3, 534–540, 2016.
  • [18] Garc ́ıa Barroso, E.R. and P 4loski, A., An approach to plane algebroid branches, Revista Matem ́atica Complutense, 28, 1, 227–252, 2015.
  • [19] Garc ́ıa Barroso, E.R. and P 4loski, A., On the intersection multiplicity of plane branches, arXiv: 1710.05346, to appear in Colloq. Math., 2017.
  • [20] Gwo ́zdziewicz, J. and P4loski,A., On the approximate roots of polynomials, Ann. Polon. Math., 60, no. 3, 199–210, 1995.
  • [21] Hirschfeld, J.W.P., Korchm ́aros, G. and Torres, F., Algebraic curves over a finite field. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, xx+696 pp., 2008.
  • [22] Jelonek, Z., Sets determining polynomial automorphism of C2, Bull. Polish Acad. Sci. Math., 37, no. 1–6, 247–250, 1989.
  • [23] Kang, M-Ch., On Abhyankar Moh’s epimorphism theorem, Amer. J. of Math., 113, 399–421, 1991.
  • [24] Kunz, E., Introduction to Plane Algebraic Curves, Translated from the 1991 German edition by Richard G. Belshoff. Birkh ̈ auser Boston, Inc., Boston, MA, xiv+293 pp., 2005.
  • [25] Moh, T.T., On analytic irreducibility at ∞of a pencil of curves, Proc. Amer. Math. Soc., 44, 22–24, 1974.
  • [26] Nagata, M., A theorem of Gutwirth, J. Math. Kyoto Univ., 11, 149–154, 1971.
  • [27] Pinkham, H., Courbes planes ayant une seule place a l’infini, S ́eminaire sur les Singularit ́es des surfaces, Centre de Math ́ematiques de l’ ́Ecole Polytechnique, Ann ́ee 1977–1978.
  • [28] P4loski, A., Introduction to the local theory of plane algebraic curves, Analytic and Algebraic Geometry, Eds. T. Krasi ́nski and S. Spodzieja. 4L ́od ́z University Press, 115–134, 2013.
  • [29] Popescu-Pampu, P., Approximate roots, Valuation theory and its applications, Vol. II (Saskatoon, SK, 1999), Fields Inst. Commun., 33, Amer. Math. Soc., Providence, RI, 285– 321, 2003.
  • [30] Reguera L ́opez, A., Semigroups and clusters at infinitiy, Algebraic geometry and singularities (La R ́abida, 1991), Progr. Math., 134, Birkh ̈auser, Basel, 339–374, 1996.
  • [31] Russell, P., Hamburger-Noether expansions and approximate roots of polynomials, Manuscripta Math., 31, no. 1–3, 25–95, 1980.
  • [32] Sathaye, A., On planar curves, Amer. J. Math., 99, no. 5, 1105–1135, 1977.
  • [33] Sathaye, A. and Stenerson, J., Plane polynomial curves, Algebraic geometry and its appli- cations, West Lafayette, IN, 1990, Springer, New York, 121–142, 1994.
  • [34] Seidenberg, A., Elements of the theory of algebraic curves, Addison-Wesley Publishing Co., Reading, Mass.- London-Don Mills, Ont. viii+216 pp., 1968.
  • [35] Smale, S., Mathematical Problems for the Next Century, Mathematical Intelligencer, vol. 20, Springer, 7–15, 1998.
  • [36] Suzuki, M., Propri ́et ́es topologiques des polynˆ omes de deux variables complexes et automor- phisms alg ́ebriques de l’espace C2, J. Math. Soc. Japan, 26, 3, 241–257, 1974.
  • [37] Suzuki, M., Affine plane curves with one place at infinity, Ann. Inst. Fourier (Grenoble), 49, no. 2, 375–404, 1999.
  • [38] van den Essen, A., Polynomial automorphisms and the jacobian conjecture, Progress in Mathematics, vol. 190. Birkh ̈auser, 2000.
  • [39] van der Kulk, W., On polynomial rings in two variables, Nieuw Arch. Wiskunde, 1, 3, 33–41, 1953.
  • [40] Xu, Y., A strong Abhyankar-Moh theorem and criterion of embedded line, J. Algebra, 409, 382–386, 2014