///////////////////////////////////
// Modifications for Theta Graph //
///////////////////////////////////


// The following contains an example for Cell7_0 combinatorial type of the modified tropical theta graphs in R^3. The curve can be visualized by means of projections. 

// We use the parameters b5, b4, b34, b2, where b3 = b34+b4, a5 = -b5^2, a4 = b4^2, a3 = b3^2, a2 = b2^2. we use this strategy since the modifications involve square roots.

// The parameters w5,w3, w34, w2 satisfy: 
// PiecesTypeIICone[1][0].Hrepresentation()
// (An inequality (1, -1, 0, 0) x + 0 >= 0,
// An inequality (0, 0, -1, 1) x + 0 >= 0,
// An inequality (-1, 2, 0, -1) x + 0 >= 0)

// We want a cancellation, so we need in(b4^2) = in(b2*b5). We aim for a subdivision containing the edge x^2y--x^3. For this, we multiply our original vector by 3 (so we replace t by t^3) and take one term of b2 to be t^(4+e) for e =1, since we need ht(x3) >2*d34/3+w2+w3+7*w5/3 and we want integer heights.

/////////////////////////////////
// Case 7_0: [0, -6, -15, -12] //
/////////////////////////////////

LIB "all.lib";
LIB "poly.lib";
LIB "tropical.lib";
LIB "elim.lib";

ring rr = (0,t), (b2,b34,b4,b5, x,y,z),dp;
poly f=y^2-x*(x-b2^2)*(x-(b4+b34)^2)*(x-b4^2)*(x+b5^2);
// f;
// b2^2*b34^2*b4^2*b5^2*x+2*b2^2*b34*b4^3*b5^2*x+b2^2*b4^4*b5^2*x+b2^2*b34^2*b4^2*x^2+2*b2^2*b34*b4^3*x^2+b2^2*b4^4*x^2-b2^2*b34^2*b5^2*x^2-2*b2^2*b34*b4*b5^2*x^2-2*b2^2*b4^2*b5^2*x^2-b34^2*b4^2*b5^2*x^2-2*b34*b4^3*b5^2*x^2-b4^4*b5^2*x^2-b2^2*b34^2*x^3-2*b2^2*b34*b4*x^3-2*b2^2*b4^2*x^3-b34^2*b4^2*x^3-2*b34*b4^3*x^3-b4^4*x^3+b2^2*b5^2*x^3+b34^2*b5^2*x^3+2*b34*b4*b5^2*x^3+2*b4^2*b5^2*x^3+b2^2*x^4+b34^2*x^4+2*b34*b4*x^4+2*b4^2*x^4-b5^2*x^4-x^5+y^2

setring(rr);
poly B5 = (1);
poly B4 = (2*t^6);
poly B34 = (t^15);
poly B2 = (4*t^12+t^13);

map P2 = rr, B2, B34, B4, B5, x, y, z;
poly ff = P2(f);

ring r = (0,t),(x,y),dp;
map newP2 = rr, 0,0,0,0,x,y,0;
poly f2 = newP2(ff);
// f2;
// -x5+(t30+t26+8t25+16t24+4t21+8t12-1)*x4+(-t56-8t55-16t54-4t47-32t46-64t45-4t42-8t38-64t37-128t36-16t33+t30+t26+8t25+4t21+8t12)*x3+(4t68+32t67+64t66+16t59+128t58+256t57-t56-8t55-16t54+16t50+128t49+256t48-4t47-32t46-64t45-4t42-8t38-64t37-128t36-16t33-16t24)*x2+y2+(4t68+32t67+64t66+16t59+128t58+256t57+16t50+128t49+256t48)*x

drawTropicalCurve(f2,"max");

///////////////////
// XZ-projection //
///////////////////

poly B5 = newP2(B5);
poly B4 = newP2(B4);
poly B34 = newP2(B34);
poly B2 = newP2(B2);
poly B3 = B34 + B4;

poly g2 = substitute(f2, y, y+B3*B4*B5*x-B5*x^2);
// g2;
// -x5+(t30+t26+8t25+16t24+4t21+8t12)*x4+(-t56-8t55-16t54-4t47-32t46-64t45-4t42-8t38-64t37-128t36-16t33+t30+t26+8t25)*x3-2*x2y+(4t68+32t67+64t66+16t59+128t58+256t57-t56-8t55-16t54+16t50+128t49+256t48-4t47-32t46-64t45-8t38-64t37-128t36)*x2+(4t21+8t12)*xy+y2+(4t68+32t67+64t66+16t59+128t58+256t57+16t50+128t49+256t48)*x

drawTropicalCurve(g2,"max");

///////////////////
// ZY-projection //
///////////////////

ring s = (0,t),(x,y,z),dp;
map P = r, x,y;
poly ff= P(f2);

poly B5 = P(B5);
poly B4 = P(B4);
poly B3 = P(B3);
poly B34 = P(B34);

ideal I = (ff,z-y+B3*B4*B5*x-B5*x^2);

poly fnox= eliminate(I,x)[1];

// Replace z by y and y by x so that the pictures are not flipped.
ring r2 = (0,t),(x,y),dp;
setring(r2);
map PP = s,0,x,y;
poly newfnox = PP(fnox);
// newfnox;
// x5-5*x4y+10*x3y2-10*x2y3+5*xy4-y5+(t60+t52+22t51+96t50+256t49+256t48-2t47-16t46-32t45+12t42-4t38-32t37-64t36+2t30+2t26+16t25+32t24)*x4+(-4t60-4t52-88t51-384t50-1024t49-1024t48+8t47+64t46+128t45-48t42+16t38+128t37+256t36-4t30-4t26-32t25-64t24-4t21-8t12)*x3y+(6t60+6t52+132t51+576t50+1536t49+1536t48-12t47-96t46-192t45+72t42-24t38-192t37-384t36+2t30+2t26+16t25+32t24+10t21+20t12+4)*x2y2+(-4t60-4t52-88t51-384t50-1024t49-1024t48+8t47+64t46+128t45-48t42+16t38+128t37+256t36-8t21-16t12-4)*xy3+(t60+t52+22t51+96t50+256t49+256t48-2t47-16t46-32t45+12t42-4t38-32t37-64t36+2t21+4t12+1)*y4+(t112+16t111+96t110+256t109+256t108-2t107-16t106-32t105+6t103+96t102+576t101+1536t100+1536t99-16t98-128t97-256t96+12t94+184t93+1152t92+3072t91+3072t90-48t89-384t88-768t87-48t84+2t81-48t80-384t79-768t78+6t77+48t76+2t73+32t72+192t71+512t70+512t69+8t68+64t67+64t66+4t64+48t63+384t62+1024t61+1025t60-16t59-128t58-256t57+2t56+16t55+16t54+t52+16t51+80t50+128t49)*x3+(-3t112-48t111-288t110-768t109-768t108+6t107+48t106+96t105-18t103-288t102-1728t101-4608t100-4608t99+48t98+384t97+768t96-36t94-552t93-3456t92-9216t91-9216t90+144t89+1152t88+2304t87+144t84-6t81+144t80+1152t79+2304t78-6t77-48t76+192t75-6t73-128t72-576t71-1536t70-1536t69+192t66-12t64-272t63-1152t62-3072t61-3075t60+48t59+384t58+768t57-2t56-16t55-112t54-3t52-60t51-240t50-384t49+4t47+32t46+64t45-24t42+8t38+64t37+128t36)*x2y+(3t112+48t111+288t110+768t109+768t108-6t107-48t106-96t105+18t103+288t102+1728t101+4608t100+4608t99-48t98-384t97-768t96+36t94+552t93+3456t92+9216t91+9216t90-144t89-1152t88-2304t87-144t84+6t81-144t80-1152t79-2304t78-288t75+6t73+144t72+576t71+1536t70+1536t69-12t68-96t67-384t66+12t64+336t63+1152t62+3072t61+3075t60-48t59-384t58-768t57+144t54+3t52+66t51+240t50+384t49-6t47-48t46-96t45+36t42-12t38-96t37-192t36)*xy2+(-t112-16t111-96t110-256t109-256t108+2t107+16t106+32t105-6t103-96t102-576t101-1536t100-1536t99+16t98+128t97+256t96-12t94-184t93-1152t92-3072t91-3072t90+48t89+384t88+768t87+48t84-2t81+48t80+384t79+768t78+96t75-2t73-48t72-192t71-512t70-512t69+4t68+32t67+128t66-4t64-112t63-384t62-1024t61-1025t60+16t59+128t58+256t57-48t54-t52-22t51-80t50-128t49+2t47+16t46+32t45-12t42+4t38+32t37+64t36)*y3+(-8t145-128t144-768t143-2048t142-2048t141+16t140+128t139+256t138-48t136-768t135-4608t134-12286t133-12256t132+320t131+1536t130+2560t129-4t128-128t127-1600t126-9216t125-24560t124-24320t123+1920t122+7168t121+10240t120-48t119-448t118-1792t117-6144t116-16336t115-15616t114+5120t113+16385t112+20496t111-112t110-1408t109-3072t108-2t107+32t106+736t105+4864t104+14342t103+16480t102+192t101-1536t100-4608t99-20t98-160t97-320t96+12t94+192t93+896t92+1024t91-1024t90-64t89-512t88-1024t87-64t80-512t79-1024t78)*x2+(16t145+256t144+1536t143+4096t142+4096t141-32t140-256t139-512t138+96t136+1536t135+9216t134+24572t133+24512t132-640t131-3072t130-5120t129+8t128+256t127+3200t126+18432t125+49120t124+48640t123-3840t122-14336t121-20480t120+80t119+768t118+3328t117+12288t116+32672t115+31264t114-10240t113-32770t112-40992t111+128t110+2048t109+4608t108+4t107-64t106-1216t105-9728t104-28684t103-32960t102-576t101+1536t100+6144t99+32t98+256t97+1280t96-24t94-368t93-1920t92-3072t91+96t89+768t88+2560t87+96t84+96t80+768t79+2048t78+192t75+128t66)*xy+(-8t145-128t144-768t143-2048t142-2048t141+16t140+128t139+256t138-48t136-768t135-4608t134-12286t133-12256t132+320t131+1536t130+2560t129-4t128-128t127-1600t126-9216t125-24560t124-24320t123+1920t122+7168t121+10240t120-40t119-384t118-1664t117-6144t116-16336t115-15632t114+5120t113+16385t112+20496t111-64t110-1024t109-2304t108-2t107+32t106+608t105+4864t104+14342t103+16480t102+288t101-768t100-3072t99-16t98-128t97-640t96+12t94+184t93+960t92+1536t91-48t89-384t88-1280t87-48t84-48t80-384t79-1024t78-96t75-64t66)*y2+(-16t166-256t165-1536t164-4096t163-4096t162+32t161+256t160+512t159-128t157-2048t156-12288t155-32768t154-32768t153+320t152+2560t151+5120t150-384t148-6144t147-36864t146-98312t145-98432t144+512t143+8192t142+18432t141+16t140-384t139-7936t138-49152t137-131120t136-131840t135-2048t134+8192t133+28672t132+128t131+768t130-2048t129-24576t128-65632t127-67072t126-6656t125-4096t124+16384t123+384t122+3072t121+6144t120-64t118-1024t117-5120t116-8192t115+512t113+4096t112+8192t111+256t104+2048t103+4096t102)*x+(16t166+256t165+1536t164+4096t163+4096t162-32t161-256t160-512t159+128t157+2048t156+12288t155+32768t154+32768t153-320t152-2560t151-5120t150+384t148+6144t147+36864t146+98312t145+98432t144-512t143-8192t142-18432t141-16t140+384t139+7936t138+49152t137+131120t136+131840t135+2048t134-8192t133-28672t132-128t131-768t130+2048t129+24576t128+65632t127+67072t126+6656t125+4096t124-16384t123-384t122-3072t121-6144t120+64t118+1024t117+5120t116+8192t115-512t113-4096t112-8192t111-256t104-2048t103-4096t102)*y

drawTropicalCurve(newfnox,"max");