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  "body": "\n## Summary\n\nThe Newton polygon of a polynomial over a valued field is a broken line built from its coefficients $a_i$, see the [[definition|D186VNHFZJ]].\nIt is the lower boundary of the [[convex hull|0FQSDC2EQ2]] of the points $(i, v(a_i))$.\nIts slopes and their lengths give the [[valuations|B85T5VKR3M]] of the roots and how many roots have each valuation.\nIt is a quick tool for factoring polynomials over $p$-adic fields and for finding [[Puiseux series|SENTYNSKSF]] solutions of algebraic equations.\n\n## Explanations\n\n### L1 human\n\nPlot one point for each coefficient: the horizontal position is the power of $X$, the height is how divisible the coefficient is by $p$.\nStretch a string under these points from the leftmost to the rightmost point and pull it tight from below.\nThe broken line you get is the Newton polygon.\nEach straight piece with slope $s$ and width $\\ell$ says: exactly $\\ell$ roots have valuation $-s$.\nFor $X^2 + X + p$ the pieces have slopes $-1$ and $0$, so one root is divisible by $p$ and the other is not.\n",
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