---
tag: SENTYNSKSF
kind: concept
name: Puiseux series
msc: [13F25, 14H20, 12J25]
sources:
  - url: https://en.wikipedia.org/wiki/Puiseux_series
same_as:
  wikidata: Q401319
  wikipedia: Puiseux series
---

## Summary

A Puiseux series is a formal Laurent series in a fractional power $X^{1/m}$ of the variable, for some positive integer $m$, see the [[definition|D5BDZZW9D4]].
Puiseux series form a field that contains the [[field of formal Laurent series|YZWYF7DXTW]] $k((X))$.
By the [[Newton–Puiseux theorem|ZWT5HGRFXX]], over an algebraically closed field of characteristic $0$ they give all roots of polynomial equations with Laurent series coefficients.
They describe the branches of a plane algebraic curve near a point.

## Explanations

### L1 human

The equation $Y^2 = X$ has no solution that is a power series in $X$, but it has the solution $Y = X^{1/2}$.
Puiseux series allow such fractional powers, as long as all exponents have a common denominator and only finitely many are negative.
An example is $X^{-1/3} + 2 + X^{1/3} + X^{2/3} + \cdots$.
Near the origin, the curve $Y^2 = X^3$ has the two branches $Y = X^{3/2}$ and $Y = -X^{3/2}$, which are Puiseux series.