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  "body": "\n## Summary\n\nA 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]].\nPuiseux series form a field that contains the [[field of formal Laurent series|YZWYF7DXTW]] $k((X))$.\nBy the [[Newton–Puiseux theorem|ZWT5HGRFXX]], over an algebraically closed field of characteristic $0$ they give all roots of polynomial equations with Laurent series coefficients.\nThey describe the branches of a plane algebraic curve near a point.\n\n## Explanations\n\n### L1 human\n\nThe equation $Y^2 = X$ has no solution that is a power series in $X$, but it has the solution $Y = X^{1/2}$.\nPuiseux series allow such fractional powers, as long as all exponents have a common denominator and only finitely many are negative.\nAn example is $X^{-1/3} + 2 + X^{1/3} + X^{2/3} + \\cdots$.\nNear 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.\n",
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