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While studying Analytic combinatorics of Flajolet and Sedgewick (to be more specific, the coefficient asymptotics of algebraic functions), I have come across the concept of Newton-Puiseux expansions. Flajolet and Sedgewick explain these series only in a sketchy way, so I have been looking for some other sources in order to gain better understanding.

However, most explanations I have found rely on quite sophisticated mathematics, such as modern algebraic geometry or Riemann surfaces. I do not doubt this is the most elegant way to approach the topic. But on the other hand, none of these have been known in the era of Newton. For this reason, I guess there should be some more elementary approach to the topic. This would be preferable for me, as my mathematical background is quite modest.

So my question is: can you recommend me a reference for a more-or-less elementary introduction to Newton-Puiseux series?

Thanks a lot.

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You can check the book "Singularities of Plane Curves" by Eduardo Casas-Alvero. If I remember correctly the first one or two chapters contain an elementary introduction to Newton-Puiseux series and the Newton method.

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