"Source: Text/Ring/Newton.text";
"Line: 1377";
"Date: Fri Sep 26 12:10:11 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Ring/Newton.text, line: 1377
// Example: H56E9 ()
print "Example: H56E9";
ei := GetEchoInput();
SetEchoInput(true);
P<x> := PuiseuxSeriesRing(Rationals());
R<y> := PolynomialRing(P);
f := (y^2 - x^3)^2 - y*x^6;
time D := DuvalPuiseuxExpansion(f, 0);
D;
time P := ParametrizationToPuiseux(D[1]);
A<a> := Parent(P[1]);
N<n> := CoefficientRing(A);
P;
time c := PuiseuxExpansion(f, 0); 
time D := DuvalPuiseuxExpansion(f, 10);
D;                                       
time P := ParametrizationToPuiseux(D[1]);
time c := PuiseuxExpansion(f, 10);       
A<a> := Parent(c[1]);
N<n> := CoefficientRing(A);
c;
A<a> := Parent(P[1]);
N<n> := CoefficientRing(A);
P;
time D := DuvalPuiseuxExpansion(f, 3);   
time P := ParametrizationToPuiseux(D[1]);
A<a> := Parent(P[1]);
N<n> := CoefficientRing(A);
P;
f := (-x^3 + x^4) - 2*x^2*y - x*y^2 + 2*x*y^4 + y^5;
time D := DuvalPuiseuxExpansion(f, 0);
D;
time P := ParametrizationToPuiseux(D[1]);
P;
Parent(P[1]);
time P := ParametrizationToPuiseux(D[2]);
A<a> := Parent(P[1]);
N<n> := CoefficientRing(A);
P;
Parent(P[1]);
N;
f := y - x^3 - x^7 - x^76 + O(x^200);
D := DuvalPuiseuxExpansion(f, 0);
D;
D := DuvalPuiseuxExpansion(f, 3);
D;
c := PuiseuxExpansion(f, 197);
c;
f := y - x^3 - x^7 - x^76;           
D := DuvalPuiseuxExpansion(f, 0);
D;
D := DuvalPuiseuxExpansion(f, 3);
D;
D := DuvalPuiseuxExpansion(f, 4);
D;                               
c := PuiseuxExpansion(f, 10); 
c;
c := PuiseuxExpansion(f, 100);
c;
c := PuiseuxExpansion(f, 200);
c;
c := PuiseuxExpansion(f, 200 : PreciseRoot := true);
c;
f := (y^2 - x^3)^2 - y*x^6;
time D := DuvalPuiseuxExpansion(f, 0);
time P := ParametrizationToPuiseux(D[1]);
A<a> := Parent(P[1]);
N<n> := CoefficientRing(A);
P;
time ExpandToPrecision(f, P[1], 20);
time D := DuvalPuiseuxExpansion(f, 5);
time P := ParametrizationToPuiseux(D[1]);
time p1 := ExpandToPrecision(f, P[1], 50);     
A<a> := Parent(p1);
N<n> := CoefficientRing(A);
p1;
time D := DuvalPuiseuxExpansion(f, 13);
time P := ParametrizationToPuiseux(D[1]);
SetEchoInput(ei);
