Newton Polygons

• Introduction
• Newton Polygons
  • Creation of Newton Polygons
    • NewtonPolygon(f): RngMPolElt → NwtnPgon
    • NewtonPolygon(f): RngUPolElt → NwtnPgon
    • NewtonPolygon(f, p): RngUPolElt, RngOrdIdl → NwtnPgon
    • NewtonPolygon(f, p): RngUPolElt, RngFunOrdIdl → NwtnPgon
    • NewtonPolygon(f, p): RngUPolElt, RngIntElt → NwtnPgon
    • NewtonPolygon(f, p): RngUPolElt, PlcFunElt → NwtnPgon
    • NewtonPolygon(C): Crv → NwtnPgon
    • NewtonPolygon(V): SeqEnum → NwtnPgon
    • NewtonPolygon(V): SetEnum → NwtnPgon
    • DefiningPoints(N): NwtnPgon → SeqEnum
    • Example: Creating Newton Polygons
  • Vertices and Faces of Polygons
    • Faces(N): NwtnPgon → SeqEnum
    • InnerFaces(N): NwtnPgon → SeqEnum
    • LowerFaces(N): NwtnPgon → SeqEnum
    • OuterFaces(N): NwtnPgon → SeqEnum
    • AllFaces(N): NwtnPgon → SeqEnum
    • Example: Faces of a Newton Polygon
    • Vertices(N): NwtnPgon → SeqEnum
    • InnerVertices(N): NwtnPgon → SeqEnum
    • LowerVertices(N): NwtnPgon → SeqEnum
    • OuterVertices(N): NwtnPgon → SeqEnum
    • AllVertices(N): NwtnPgon → SeqEnum
    • Example: Vertices of a Newton Polygon
    • EndVertices(F): NwtnPgonFace → SeqEnum
    • FacesContaining(N,p): NwtnPgon, Tup → SeqEnum
    • Example: End Vertices and Containing Faces
    • GradientVector(F): NwtnPgonFace → Tup
    • GradientVectors(N): NwtnPgon → [ Tup ]
    • Weight(F): NwtnPgonFace → RngIntElt
    • Slopes(N): NwtnPgon → SeqEnum
    • InnerSlopes(N): NwtnPgon → SeqEnum
    • LowerSlopes(N): NwtnPgon → SeqEnum
    • AllSlopes(N): NwtnPgon → SeqEnum
    • Example: Gradient Vector and Weight of a Face
  • Tests for Points and Faces
    • IsFace(N, F): NwtnPgon, Tup → BoolElt
    • IsVertex(N, p): NwtnPgon, Tup → BoolElt
    • IsInterior(N,p): NwtnPgon, Tup → BoolElt
    • IsBoundary(N, p): NwtnPgon, Tup → BoolElt
    • IsPoint(N,p): NwtnPgon, Tup → BoolElt
• Polynomials Associated with Newton Polygons
  • HasPolynomial(N): NwtnPgon → BoolElt
  • Polynomial(N): NwtnPgon → RngElt
  • ParentRing(N): NwtnPgon → Rng
  • IsNewtonPolygonOf(N, f): NwtnPgon, RngElt → BoolElt
  • FaceFunction(F): NwtnPgonFace → RngElt
  • IsDegenerate(F): NwtnPgonFace → BoolElt
  • IsDegenerate(N): NwtnPgon → BoolElt
  • IsNondegenerate(F): NwtnPgonFace → BoolElt
  • IsNondegenerate(N): NwtnPgon → BoolElt
• Finding Valuations of Roots of Polynomials from Newton Polygons
  • ValuationsOfRoots(f): RngUPolElt → [ < RngIntElt, RngIntElt > ]
  • ValuationsOfRoots(f, p): RngUPolElt, RngIntElt → [ < RngIntElt, RngIntElt > ]
  • ValuationsOfRoots(f, p): RngUPolElt, RngOrdIdl → [ < RngIntElt, RngIntElt > ]
  • ValuationsOfRoots(f, p): RngUPolElt, PlcFunElt → [ < RngIntElt, RngIntElt > ]
• Using Newton Polygons to Find Roots of Polynomials over Series Rings
  • SetVerbose("Newton", v): MonStgElt, RngIntElt
  • Operations not associated with Duval’s Algorithm
    • PuiseuxExpansion(f, n): RngUPolElt, RngIntElt → SeqEnum[RngSerPuisElt]
    • ExpandToPrecision(f, c, n): RngUPolElt, RngSerElt, RngIntElt → RngSerElt
    • ImplicitFunction(f, d, n): RngUPolElt, RngIntElt, RngIntElt → RngSerElt
    • Example: Puiseux Expansions to Higher Precision
    • IsPartialRoot(f, c): RngUPolElt, RngSerElt → BoolElt
    • IsUniquePartialRoot(f, c): RngUPolElt, RngSerElt → BoolElt
    • Example: Checking That a Partial Root Can Be Expanded
    • PuiseuxExponents(p): RngSerElt → SeqEnum
    • PuiseuxExponentsCommon(p, q): RngSerElt, RngSerElt → SeqEnum
    • Example: Exponents of Puiseux Expansions
  • Operations associated with Duval’s algorithm
    • DuvalPuiseuxExpansion(f, n): RngUPolElt, RngIntElt → SeqEnum
    • ParametrizationToPuiseux(T): Tup → SeqEnum
    • PuiseuxToParametrization(S): RngSerElt → Tup
    • Example: Duval's Algorithm for Puiseux Expansions
  • Roots of Polynomials
    • Roots(f): RngUPolElt → [<RngSerElt, RngIntElt>]
    • Roots(f, n): RngUPolElt, RngIntElt → [<RngSerElt, RngIntElt>]
    • HasRoot(f): RngUPolElt → BoolElt, RngSerElt
    • Example: Roots of Polynomials over Puiseux Series
• Bibliography