Sampling Galois groupsΒΆ

This page contains the code for the experiments in Section 5 for Galois groups:

GaloisGroups.m

powers := [[0, 0, 0, 3],
           [0, 0, 1, 2],
           [0, 0, 2, 1],
           [0, 0, 3, 0],
       [0, 1, 0, 2],
       [0, 1, 1, 1],
           [0, 1, 2, 0],
       [0, 2, 0, 1],
       [0, 2, 1, 0],
       [0, 3, 0, 0],
       [1, 0, 0, 2],
           [1, 0, 1, 1],
           [1, 0, 2, 0],
           [1, 1, 0, 1],
           [1, 1, 1, 0],
           [1, 2, 0, 0],
           [2, 0, 0, 1],
           [2, 0, 1, 0],
           [2, 1, 0, 0],
           [3, 0, 0, 0]];

function randomSmoothCubicSurface(N,p)

        _<x,y,z,w> := PolynomialRing(Rationals(),4);
        P3<x,y,z,w> := ProjectiveSpace(Rationals(),3);

        listCoef := [0..p^(N+1)-1];

        bool := true;
        while bool do

                randSurface := 0*x;
                for part in powers do
                        coef := Random(listCoef);
                        monomial := x^part[1] * y^part[2] * z^part[3] * w^part[4];
                        randSurface := randSurface + coef * monomial;
                end for;

                S := Surface(P3,randSurface);
                bool := IsSingular(S);

        end while;

        return randSurface;

end function;



function galoisGroup(surface,p)

        f := surface;

        QQabcd<a,b,c,d> := PolynomialRing(Rationals(),4);
        _<s,t> := PolynomialRing(QQabcd,2);

        param := [a*s+b*t,c*s+d*t,s,t];

        pt0 := [1,0];
        pt1 := [1,1];
        ptmin1 := [1,-1];
        ptinf := [0,1];

        g0 := Evaluate(f,[Evaluate(param[i],pt0) : i in [1..4]]);
        g1 := Evaluate(f,[Evaluate(param[i],pt1) : i in [1..4]]);
        gmin1 := Evaluate(f,[Evaluate(param[i],ptmin1) : i in [1..4]]);
        ginf := Evaluate(f,[Evaluate(param[i],ptinf) : i in [1..4]]);

        I := Ideal([g0,g1,gmin1,ginf]);
        GB := GroebnerBasis(I);
        g := GB[#GB];

        K := pAdicField(p,300);

        newcoeff := [];
        for i in [0..27] do
                Append(~newcoeff,K!Rationals()!Coefficient(g,4,i));
        end for;

        _<x> := PolynomialRing(K);
        gK := 0*x;
        for i in [1..28] do
        gK := gK + newcoeff[i]*x^(i-1);
        end for;

        try G := GaloisGroup(gK);
        catch e return -1;
        end try;

        return G;

end function;



function galoisGroups(N,p,M)
        abelian_groups := [];
        nonabelian_groups := [];

        for i in [1..M] do
                print i;

                f := randomSmoothCubicSurface(N,p);
                group := galoisGroup(f,p);

                if Type(group) eq GrpPerm then
                        if IsAbelian(group) then
                                Append(~abelian_groups,group);
                        else
                                Append(~nonabelian_groups,group);
                        end if;
                end if;

        end for;

        return [abelian_groups,nonabelian_groups];

end function;