Math 2.0

Field theory and polynomials Definition

Newton polygon

Builds on ValuationConvex hull

Definition

Let KK be a field with a valuation v ⁣:K→ℝ∪{∞}v \colon K \to \R \cup \set{\infty}. Let f=a0+a1X+⋯+anXn∈K[X]f = a_0 + a_1 X + \cdots + a_n X^n \in K[X] with n≥1n \ge 1 and a0an≠0a_0 a_n \ne 0. Let S={(i,v(ai)):0≤i≤n, ai≠0}⊆ℝ2S = \set{(i, v(a_i)) : 0 \le i \le n,\ a_i \ne 0} \subseteq \R^2. The Newton polygon of ff is the lower boundary of the convex hull of SS. It is the graph of the largest convex function φ ⁣:[0,n]→ℝ\varphi \colon [0, n] \to \R with φ(i)≤v(ai)\varphi(i) \le v(a_i) for every (i,v(ai))∈S(i, v(a_i)) \in S.

It is a broken line from (0,v(a0))(0, v(a_0)) to (n,v(an))(n, v(a_n)) whose vertices lie in SS and whose slopes increase from left to right.

Remarks

Main property: let LL be a splitting field of ff over KK and ww an extension of vv to LL. If the Newton polygon has a segment of slope ss and horizontal length ℓ\ell, then exactly ℓ\ell roots of ff in LL have ww-value −s-s, counted with multiplicity. Example: over ℚp\Q_p, take f=X2+X+pf = X^2 + X + p. The points are (0,1)(0, 1), (1,0)(1, 0), (2,0)(2, 0), and the lower boundary has a segment of slope −1-1 from (0,1)(0, 1) to (1,0)(1, 0) and a segment of slope 00 from (1,0)(1, 0) to (2,0)(2, 0). So one root has valuation 11 and one has valuation 00; this agrees with the product of the roots being pp and their sum being −1-1. Non-example: for f=1+pX+X2f = 1 + pX + X^2 the points are (0,0)(0, 0), (1,1)(1, 1), (2,0)(2, 0). The broken line through all three points is not the Newton polygon, because it is not the graph of a convex function. The point (1,1)(1, 1) lies above the segment from (0,0)(0, 0) to (2,0)(2, 0), and this segment is the whole Newton polygon. So both roots have valuation 00.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Affine combination, concept, no evidence levelAffine combinationconceptBinary operation, definition, no evidence levelBinary operationdefinitionConvex set, concept, no evidence levelConvex setconceptGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionConvex hull, definition, no evidence levelConvex hulldefinitionRing, definition, no evidence levelRingdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionField, definition, no evidence levelFielddefinitionValuation, definition, no evidence levelValuationdefinitionNewton polygon, definition, no evidence levelNewton polygondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Affine combination concept
  2. Binary operation definition
  3. Convex set concept
  4. Group concept
  5. Commutativity definition
  6. Convex hull definition
  7. Ring definition
  8. Abelian group definition
  9. Commutative ring definition
  10. Unit of a ring definition
  11. Field definition
  12. Valuation definition
  13. Newton polygon definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Newton_polygon
About this page
TagD186VNHFZJ
URIm2://D186VNHFZJ
Kinddefinition
WritingW0 stub
MSC12J25, 11S05
DefinesNewton polygon
UsesValuation, Convex hull
Changed2 November 2026

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