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Take an example polynomial $f(x, y) = y^2 x + y^3 - x^2$. A solution to $f(x,y)=0$ exists with Puiseux series given by $y(x) = x^{2/3} - x/3 + x^{4/3}/9+\cdots$. I got this by having Mathematica directly solve $f=0$ for $y$ and then perform a Puiseux series expansion for me about $x=0$.

However, since a direct solution to $f=0$ is obviously not available for arbitrary $f$, what I'd like is a method to determine just the lowest-order term of the expansion of $y(x)$ about $x=0$ for arbitrary $f(x,y)$. In the example above, this term would be $x^{2/3}$.


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2 Answers 2

up vote 3 down vote accepted

Finding such solutions was in fact Newton's original motivation for inventing "Newton polygons". There's a nice exposition of this in Chapter 2 of "The Implicit Function Theorem: History, Theory, and Applications" by Steve Krantz and Hal Parks.

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The magic words are "Newton Polygon". (there is a vast literature on the subject, the wikipedia article is one place to start).

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