Newton–Puiseux theorem

Remarks

Proof idea (Newton's method): let $F(Y)$ be a polynomial with coefficients in $P$. A slope of the Newton polygon of $F$ for the $X$-adic valuation gives a possible leading exponent $\gamma$ of a root. A root $c \ne 0$ in $k$ of an associated polynomial gives the leading term $c X^\gamma$. One then substitutes $Y = X^\gamma(c + Y_1)$ and repeats. In characteristic $0$ one shows that the denominators of the exponents stop growing, so the process produces a Puiseux series. A second proof shows that every finite extension of $k((X))$ of degree $n$ is $k((X^{1/n}))$. Example: the roots of $Y^2 - X^3$ are $\pm X^{3/2}$, and the roots of $Y^2 - X^2(1 + X)$ are $\pm X(1 + X)^{1/2}$, with $(1 + X)^{1/2}$ given by the binomial series. The characteristic $0$ hypothesis is necessary: over an algebraic closure of $\F_p$, the polynomial $Y^p - Y - X^{-1}$ has no root in the field of Puiseux series.

Going deeper

For a student

Geometrically, the theorem parametrizes each branch of a plane algebraic curve $F(X, Y) = 0$ through a point by $X = t^m$, $Y =$ a power series in $t$. The exponents of the Puiseux expansion are the start of the classification of plane curve singularities. Over $\C$ the Puiseux series that arise from polynomials converge near $0$, and they describe the branches analytically. The Newton polygon is the computational tool, and Henselian fields give the general frame for lifting approximate roots.