// This is a Singular file that shows that there is no such solutions of the
// Ernst-Maxwell equations that 

// \varphi=O(|z-z0|) with non-zero gradient at z0 and E=O(|z-z0|^4)

timer=1;

option(prot);

int tt = timer;

int k=5;

int n=2;

ring R = 0,(a(6..9),b(6..9),c(2..9),d(2..9),rho0,w,z),dp;

poly E;

E=0;

poly F;

F=0;

// Taylor expansion of \varphi

poly Phi;

Phi=z+c(2)*w+c(3)*z^2+c(4)*w^2+c(5)*z*w+c(6)*z^3+c(7)*z^2*w+c(8)*z*w^2+c(9)*w^3;

poly Alpha;

Alpha=w+d(2)*z+d(3)*w^2+d(4)*z^2+d(5)*w*z+d(6)*w^3+d(7)*w^2*z+d(8)*w*z^2+d(9)*z^3;

// Ernst-Maxwell equations 
// (only the second equation is nontrivial in this order)

poly EM1 = (E+F+2*Alpha*Phi)*(2*(2*rho0+z+w)*diff(diff(E,w),z)+diff(E,z)+diff(E,w))-4*(2*rho0+z+w)*(diff(E,z)*diff(E,w)+Alpha*(diff(E,z)*diff(Phi,w)+diff(E,w)*diff(Phi,z)));

poly EM1c = (E+F+2*Alpha*Phi)*(2*(2*rho0+z+w)*diff(diff(F,z),w)+diff(F,w)+diff(F,z))-4*(2*rho0+z+w)*(diff(F,w)*diff(F,z)+Phi*(diff(F,w)*diff(Alpha,z)+diff(F,z)*diff(Alpha,w)));

poly EM2 = Phi*(2*(2*rho0+z+w)*diff(diff(Phi,w),z)+diff(Phi,z)+diff(Phi,w))-2*(2*rho0+z+w)*(2*diff(Phi,z)*diff(Phi,w)); 

poly EM2c = Alpha*(2*(2*rho0+z+w)*diff(diff(Alpha,z),w)+diff(Alpha,w)+diff(Alpha,z))-2*(2*rho0+z+w)*(2*diff(Alpha,w)*diff(Alpha,z));

poly f=-(E+F+2*Alpha*Phi)/2;

intvec v=0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1;

EM1=jet(EM1,n,v);
EM1c=jet(EM1c,n,v);
EM2=jet(EM2,n,v);
EM2c=jet(EM2c,n,v);

EM1;

matrix C1 = coef(EM1,zw);

matrix C1c = coef(EM1c,zw);

matrix C2 = coef(EM2,zw);

matrix C2c = coef(EM2c,zw);

ideal C;

ideal Co = C1[2,1..1]; 

C= ideal(Co);

Co = C2[2,1..6];

C=C + ideal(Co);

Co = C1c[2,1..1];

C=C + ideal(Co);

Co = C2c[2,1..6];

C=C + ideal(Co);

 ring R2 = 0,(a(6..9),b(6..9),c(2..9),d(2..9),rho0),dp;

 ideal C = imap(R,C);

"-----------------C-----------------------";
C;

// ideal gC = groebner(C);

// "dimension:",dim(gC);

//size(gC);

//gC;

 LIB "primdec.lib";

 list mpr = minAssGTZ(C); 

"";
"used time:",timer-tt, "sec";
"used memory:", memory(2) / (1024*1024), "MB";
""; 

// 0=1, hence there are no solutions!

mpr;

"test C:";

int l;
int i;
for (i=1; i<=size(mpr); i++)
{
l= l+size(reduce(C,std(mpr[i])));
}
l;