Magma V2.16-8 


###############################################
Degeneration Type I
###############################################

%% See notations of Theorem 3.1


%% The first program checks that the intermediate extension L/K
%% is weakly ramified, see Step D.

% magma code :

q2:=pAdicField(2,100);
q2x<x>:=PolynomialAlgebra(q2);
k<t>:=TotallyRamifiedExtension(q2,x^5+2);
ky<y>:=PolynomialAlgebra(k);
L:=y^16-2^6*(1+y^4+y^5)^3*(1+y);
F,s,C:=Factorization(L:  Extensions := true);
L1:=C[1]`Extension;
Valuation(Discriminant(IntegerRing(L1),IntegerRing(k)));


%% The second program checks that the filtration of the wild
%% monodromy extension is the expected one. 

% maple code :
restart;
f:=1+y^4+y^5;
R:=subs(y=z,resultant(y^16-2^6*(1+y^4+y^5)^3*(1+y),z^2-f,y)):
fd:=fopen(foo,WRITE);
fprintf(fd,cat(cat("R:=",convert(R,string)),";"));
fclose(fd);

% magma code :
q2:=pAdicField(2,1000);
q2x<x>:=PolynomialAlgebra(q2);
k<t>:=TotallyRamifiedExtension(q2,x^5+2);
kz<z>:=PolynomialAlgebra(k);
load "foo";
F,s,C:=Factorization(R: Extensions := true);
L1:=C[1]`Extension;
Valuation(Discriminant(IntegerRing(L1),IntegerRing(k)));






###############################################
Degeneration Type II
###############################################

%% See notations of Proposition 4.1.

%% One checks that T_f factors in a product of two irreducible polynomials T_{1,f} and T_{2,f} 
%% over K and that T_{2,f} is irreducible over the decomposition field of T_{1,f}.

% magma code :
q2:= pAdicField(2,8);
q2x<x>:=PolynomialRing(q2);
k<pi>:=TotallyRamifiedExtension(q2,x^15-2);
K<rho>:=UnramifiedExtension(k,8);
Ky<y>:=PolynomialRing(K);
b3:=1;
b2:=pi^9;
b4:=pi^6;
T:=(2*b2*y+3*b3*y^2+4*b4*y^3+5*y^4)^2-4*(1+b2*y^2+b3*y^3+b4*y^4+y^5)*(b2+3*b3*y+6*b4*y^2+10*y^3);
F,a,A:=Factorization(T: Extensions:= true);
Degree(F[1][1]);Degree(F[2][1]);
L1:=A[1]`Extension;
L1Y<Y>:=PolynomialAlgebra(L1);
TY:=L1Y!Eltseq(T);
G:=Factorization(TY);
G[1][2];G[2][2];G[3][2];G[4][2];G[5][2];
Degree(G[1][1]);Degree(G[2][1]);Degree(G[3][1]);Degree(G[4][1]);Degree(G[5][1]);



%% The following program computes the ramification filtrations of the 
%% subextensions K_1/K and K_2/K of the monodromy extension.

% magma code :
q2 := pAdicField(2,32);
q2x<x> := PolynomialRing(q2);
k<pi> := TotallyRamifiedExtension(q2,x^15-2);
K<rho> := UnramifiedExtension(k,8);
Ky<y> :=  PolynomialRing(K);
b3:=1;
b2:=pi^9;
b4:=pi^6;
T:= (2*b2*y+3*b3*y^2+4*b4*y^3+5*y^4)^2-4*(1+b2*y^2+b3*y^3+b4*y^4+y^5)*(b2+3*b3*y+6*b4*y^2+10*y^3);
F,a,A := Factorization(T: Extensions:= true);
L1 := A[1]`Extension;
L1Y<Y>:=PolynomialAlgebra(L1);
TY:=L1Y!Eltseq(T);
R1:=Roots(TY);
r:=R1[1][1];
P:=Y^2-(1+b2*r^2+b3*r^3+b4*r^4+r^5);
G,b,B:=Factorization(P : Extensions :=true);
L3:=B[1]`Extension;
Valuation(Discriminant(IntegerRing(L1),IntegerRing(K)));
Valuation(Discriminant(IntegerRing(L3),IntegerRing(L1)));

q2 := pAdicField(2,32);
q2x<x> := PolynomialRing(q2);
k<pi> := TotallyRamifiedExtension(q2,x^15-2);
K<rho> := UnramifiedExtension(k,8);
Ky<y> :=  PolynomialRing(K);
b3:=1;
b2:=pi^9;
b4:=pi^6;
T:= (2*b2*y+3*b3*y^2+4*b4*y^3+5*y^4)^2-4*(1+b2*y^2+b3*y^3+b4*y^4+y^5)*(b2+3*b3*y+6*b4*y^2+10*y^3);
F,a,A := Factorization(T: Extensions:= true);
L2 := A[2]`Extension;
L2Y<Y>:=PolynomialAlgebra(L2);
TY:=L2Y!Eltseq(T);
R1:=Roots(TY);
r:=R1[1][1];
P:=Y^2-(1+b2*r^2+b3*r^3+b4*r^4+r^5);
G,b,B:=Factorization(P : Extensions :=true);
L3:=B[1]`Extension;
Valuation(Discriminant(IntegerRing(L2),IntegerRing(K)));
Valuation(Discriminant(IntegerRing(L3),IntegerRing(L2)));



###############################################
Degeneration Type III
###############################################

%% See notations of Proposition 4.2.

%% The next two programs compute T_f and Delta
%% then describe their Newton polygon.

% maple code :
restart:
s0:=1+a^3*x^2+a^6*x^3+x^5;
s1:=2*a^3*x+3*a^6*x^2+5*x^4;
s2:=a^3+3*a^6*x+10*x^3;
T:=subs(x=y,collect(expand(s1^2-4*s0*s2),x));
Delta:=expand((subs(y=Y+r,T)-subs(y=r,T))/Y);
fd:=fopen(foo1,WRITE);
fprintf(fd,cat(cat("Delta:=",convert(Delta,string)),";"));
fclose(fd);
fd:=fopen(foo2,WRITE);
fprintf(fd,cat(cat("T:=",convert(T,string)),";"));
fclose(fd);

% magma code :
q2:=pAdicField(2,100);
q2x<x>:=PolynomialAlgebra(q2);
k<a>:=TotallyRamifiedExtension(q2,x^9-2);
ky<y>:=PolynomialRing(k);
load "foo2";
Slopes(NewtonPolygon(T));
F,s,C:=Factorization(T:  Extensions := true);
L1:=C[1]`Extension;
L1Y<Y>:=PolynomialAlgebra(L1);
T1Y:=L1Y!Eltseq(T);
r:=Roots(T1Y)[1][1];
load "foo1";
Slopes(NewtonPolygon(Delta));

%% One checks the announced decomposition of T_f in irreducible
%% factors over K(y_1) and that T_2 totally decomposes over K(y_1,y_5).

% magma code :
q2:=pAdicField(2,100);
q2x<x>:=PolynomialAlgebra(q2);
k<a>:=TotallyRamifiedExtension(q2,x^9-2);
K<rho>:=UnramifiedExtension(k,8);
Ky<y>:=PolynomialRing(K);
load "foo2";
IsIrreducible(T);
F,s,C:=Factorization(T:  Extensions := true);
L1:=C[1]`Extension;
L1Y<Y>:=PolynomialAlgebra(L1);
T1Y:=L1Y!Eltseq(T);
Fa:=Factorization(T1Y);
Degree(Fa[1][1]);
Degree(Fa[2][1]);
Degree(Fa[3][1]);
Degree(Fa[4][1]);
Degree(Fa[5][1]);
T2:=Fa[5][1];
G,t,D:=Factorization(T2:  Extensions := true);
L2:=D[1]`Extension;
L2Z<Z>:=PolynomialRing(L2);
T2Z:=L2Z!Eltseq(T2);
R2:=Roots(T2Z);
R2[1][2];R2[2][2];R2[3][2];R2[4][2];