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, PlcFunElt -> NwtnPgon NewtonPolygon(C) : Crv -> NwtnPgon NewtonPolygon(V) : SeqEnum -> NwtnPgon DefiningPoints(N) : NwtnPgon -> SeqEnum Example Newton_create-ex (H41E1)
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 Newton_faces-ex (H41E2) Vertices(N) : NwtnPgon -> SeqEnum InnerVertices(N) : NwtnPgon -> SeqEnum LowerVertices(N) : NwtnPgon -> SeqEnum OuterVertices(N) : NwtnPgon -> SeqEnum AllVertices(N) : NwtnPgon -> SeqEnum Example Newton_vertices-ex (H41E3) EndVertices(F) : NwtnPgonFace -> SeqEnum FacesContaining(N,p) : NwtnPgon,Tup -> SeqEnum Example Newton_sp-vertices-ex (H41E4) GradientVector(F) : NwtnPgonFace -> Tup Weight(F) : NwtnPgonFace -> RngIntElt Slopes(N) : NwtnPgon -> SeqEnum InnerSlopes(N) : NwtnPgon -> SeqEnum Example Newton_grad-ex (H41E5)
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 FaceFunction(F) : NwtnPgonFace -> RngElt
Finding Valuations of Roots of Polynomials from Newton Polygons ValuationsOfRoots(f) : RngUPolElt -> [ < RngIntElt, RngIntElt > ] ValuationsOfRoots(f, p) : RngUPolElt, RngIntElt -> [ < 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 Newton_poly-ops-ex (H41E6) IsPartialRoot(f, c) : RngUPolElt, RngSerElt -> BoolElt IsUniquePartialRoot(f, c) : RngUPolElt, RngSerElt -> BoolElt Example Newton_pol-is (H41E7) PuiseuxExponents(p) : RngSerElt -> SeqEnum PuiseuxExponentsCommon(p, q) : RngSerElt, RngSerElt -> SeqEnum Example Newton_exps (H41E8)
Operations associated with Duval's algorithm DuvalPuiseuxExpansion(f, n) : RngUPolElt, RngIntElt -> SeqEnum ParametrizationToPuiseux(T) : Tup -> SeqEnum PuiseuxToParametrization(S) : RngSerElt -> Tup Example Newton_duval-ex (H41E9)
Roots of Polynomials Roots(f) : RngUPolElt -> [<RngSerElt, RngIntElt>] HasRoot(f) : RngUPolElt -> BoolElt, RngSerElt
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