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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Analytic equations and singularities of plane curves


Author: John J. Wavrik
Journal: Trans. Amer. Math. Soc. 245 (1978), 409-417
MSC: Primary 14D15
DOI: https://doi.org/10.1090/S0002-9947-1978-0511419-5
MathSciNet review: 511419
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Abstract: Theorems (Artin, Wavrik) exist which show that sufficiently good approximate (power series) solutions to a system of analytic equations may be approximated by convergent solutions. This paper considers the problem of explicity determining the order, $ \beta $, to which an approximate solution must solve the system of equations.

The paper deals with the case of one equation, $ f(x,y) = 0$, in two variables. It is shown how $ \beta $ depends on the singularities of the curve $ f(x,y) = 0$. A method for obtaining the minimal $ \beta $ is given. A rapid way of finding $ \beta $ using the Newton Polygon for f applies in special cases.


References [Enhancements On Off] (What's this?)

  • [1] M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23–58. MR 0268188
  • [2] Julian Lowell Coolidge, A treatise on algebraic plane curves, Dover Publications, Inc., New York, 1959. MR 0120551
  • [3] William Fulton, Algebraic curves. An introduction to algebraic geometry, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes written with the collaboration of Richard Weiss; Mathematics Lecture Notes Series. MR 0313252
  • [4] H. T. Kung and J. F. Traub, All algebraic functions can be computed fast, Analyse et contrôle de systèmes (Papers, IRIA Sem., Rocquencourt, 1977), IRIA, Rocquencourt, 1977, pp. 133–172. MR 525187
  • [5] Jean-Claude Tougeron, Idéaux de fonctions différentiables, Springer-Verlag, Berlin-New York, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71. MR 0440598
  • [6] Robert J. Walker, Algebraic Curves, Princeton Mathematical Series, vol. 13, Princeton University Press, Princeton, N. J., 1950. MR 0033083
  • [7] John J. Wavrik, A theorem on solutions of analytic equations with applications to deformations of complex structures, Math. Ann. 216 (1975), no. 2, 127–142. MR 0387649, https://doi.org/10.1007/BF01432540

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0511419-5
Keywords: Analytic equations, plane curve singularities, power series, algebraic functions
Article copyright: © Copyright 1978 American Mathematical Society

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