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 | References | Similar Articles | Additional Information

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, , to which an approximate solution must solve the system of equations.

The paper deals with the case of one equation, , in two variables. It is shown how depends on the singularities of the curve . A method for obtaining the minimal is given. A rapid way of finding using the Newton Polygon for *f* applies in special cases.

**[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