{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $K$ be a field with a [[valuation|SVY882HKSJ]] $v \\colon K \\to \\R \\cup \\set{\\infty}$.\nLet $f = a_0 + a_1 X + \\cdots + a_n X^n \\in K[X]$ with $n \\ge 1$ and $a_0 a_n \\ne 0$.\nLet $S = \\set{(i, v(a_i)) : 0 \\le i \\le n,\\ a_i \\ne 0} \\subseteq \\R^2$.\nThe *Newton polygon* of $f$ is the lower boundary of the [[convex hull|0TCHAS04WK]] of $S$.\nIt is the graph of the largest convex function $\\varphi \\colon [0, n] \\to \\R$ with $\\varphi(i) \\le v(a_i)$ for every $(i, v(a_i)) \\in S$.\n\nIt is a broken line from $(0, v(a_0))$ to $(n, v(a_n))$ whose vertices lie in $S$ and whose slopes increase from left to right.\n\n## Remarks\n\nMain property: let $L$ be a splitting field of $f$ over $K$ and $w$ an extension of $v$ to $L$.\nIf the Newton polygon has a segment of slope $s$ and horizontal length $\\ell$, then exactly $\\ell$ roots of $f$ in $L$ have $w$-value $-s$, counted with multiplicity.\nExample: over $\\Q_p$, take $f = X^2 + X + p$.\nThe points are $(0, 1)$, $(1, 0)$, $(2, 0)$, and the lower boundary has a segment of slope $-1$ from $(0, 1)$ to $(1, 0)$ and a segment of slope $0$ from $(1, 0)$ to $(2, 0)$.\nSo one root has valuation $1$ and one has valuation $0$; this agrees with the product of the roots being $p$ and their sum being $-1$.\nNon-example: for $f = 1 + pX + X^2$ the points are $(0, 0)$, $(1, 1)$, $(2, 0)$.\nThe broken line through all three points is not the Newton polygon, because it is not the graph of a convex function.\nThe point $(1, 1)$ lies above the segment from $(0, 0)$ to $(2, 0)$, and this segment is the whole Newton polygon.\nSo both roots have valuation $0$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "H7QNCE2QZJ",
    "kind": "definition",
    "msc": [
      "12J25",
      "11S05"
    ],
    "name": "Newton polygon",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Newton_polygon"
      }
    ],
    "tag": "D186VNHFZJ",
    "uses": [
      "SVY882HKSJ",
      "0TCHAS04WK"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/D186VNHFZJ/",
  "links": [
    {
      "field": "defines",
      "target": "H7QNCE2QZJ"
    },
    {
      "field": "uses",
      "target": "SVY882HKSJ"
    },
    {
      "field": "uses",
      "target": "0TCHAS04WK"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/D186VNHFZJ.md",
  "msc": [
    "12J25",
    "11S05"
  ],
  "name": "Newton polygon",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D18/D186VNHFZJ.md",
  "status": null,
  "tag": "D186VNHFZJ",
  "uri": "m2://D186VNHFZJ",
  "url": "https://math2.tamnd.com/wiki/newton-polygon-definition/",
  "writing": {
    "tag": "D186VNHFZJ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}