Analytic equations and singularities of plane curves

Wavrik, John J.

doi:10.1090/S0002-9947-1978-0511419-5

Analytic equations and singularities of plane curves
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by John J. Wavrik PDF

Trans. Amer. Math. Soc. 245 (1978), 409-417 Request permission

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.

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