option(none);
// option(prot);
// option(mem);

LIB "elim.lib";
LIB "solve.lib";
LIB "primdec.lib";

print("");
print("FERMAT MIRROR K3 AS LEFSCHETZ FIBRATION (I)");
print(""); 
print("This program carries out the first step in the mirror computation,");
print("Namely, representing the 4 Lagrangian spheres as matching cycles.");
print("We use the auxiliary holomorphic function");
print("");
print("1/4 (c_0 u_0 + c_1 u_1 + c_2 u_2 + c_3 u_3)");
print("");
print("with c_k roots of unity EXCEPT c_0, where we perturb from 1 to 5/4.");

// the critical locus is given by points u_0^{-1} = s+c_0*t, etc.
// such that u_0 \dots u_3 = 1. If we write prod = (s+c_0*t)...
// (s+c_k*t), then the map from the critical locus down to \C^4
// is 1/4 times the derivatives of prod.

ring space = 0,(s,t),dp;
poly prod = (s-5/4*t)*(s+t)*(s^2+t^2);
poly derivs = diff(prod,s);
poly derivt = diff(prod,t);

ring graphspace = 0,(s,t,z,w),dp;

ideal graph = imap(space,prod)-1, 
              4*z-imap(space,derivs), 
              4*w-imap(space,derivt);  

print("");
print("The downstairs branch curve is:");
poly locus = elim1(graph,st)[1];
short = 0; factorize(locus);

ring basespace = 0,(z,w),dp;

poly locus2 = imap(graphspace,locus);
ideal tangencies = locus2,diff(locus2,w);

print("");
print("The real and fake critical values z are the zeros of (reduced):");
ideal critred = radical(elim1(tangencies,w));
short = 0; critred;

poly critz = 1;
for (int k=1; k <= size(critred); k=k+1){
  critz = critz * critred[k];
};

ring cc = (complex,30,i),z,dp;
poly tosolve = imap(basespace,critz);
short = 0; tosolve;

print("");
print("Numerically,");
short = 0; laguerre_solve(tosolve);

setring basespace;
poly locus3 = subst(locus2,z,0);

ring cc2 = (complex,30,i),w,dp;
poly locus4 = imap(basespace,locus3);

print("");
print("If we restrict to the Fermat K3 fibre (z = 0), then");
print("the w-values of the critical points are the zeros of:");
short = 0; locus4;

print("");
print("Numerically,");
short = 0; laguerre_solve(locus4);

option(prompt);