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

// The following contains an example for Cell6 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: 
// PiecesTypeIICone3[1][2].Hrepresentation()
// (An equation (0, 0, 1, -1) x + 0 == 0,
//  An inequality (1, -2, 0, 1) x + 0 >= 0,
//  An inequality (0, 1, 0, -1) x + 0 >= 0)

// We want a cancellation, so we need in(b34^2) = -in(b2^2). For this, we replace a2 with -b2^2.

////////////////////////////////////
// Case 6: w2+w5 > 2w3 , w34 = w2 //
////////////////////////////////////


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+t^2);
poly B4 = (4*t^2+t^5);
poly B34 = (t^3+t^10);
poly B2 = (t^3+t^6);

map P2 = rr, B2, B34, B4, B5, x, y, z;
poly ff = P2(f);
w
ring r = (0,t),(x,y),dp;
map newP2 = rr, 0,0,0,0,x,y,0;
poly f2 = newP2(ff);
// f2;
// -x5+(t20+2t15+2t13+7t12+2t10-2t9+2t8+16t7+8t5+31t4-2t2-1)*x4+(t32-t30+2t29-6t27+t26-3t24-2t23-16t22+18t21-14t20-72t19+7t18-30t17-56t16-16t15-62t14+72t13-82t12-236t11+22t10-90t9-222t8+32t7+64t6+8t5+32t4)*x3+(-t42-10t39-2t37-32t36-2t35-27t34-38t33-22t32-148t31-18t30-94t29-347t28-36t27-234t26-348t25-201t24-536t23-202t22-510t21-928t20-208t19-549t18-776t17-330t16-254t15-508t14-552t13-319t12-504t11-496t10-128t9-256t8)*x2+y2+(-t46-2t44-10t43-t42-22t41-33t40-16t39-94t38-46t37-110t36-228t35-88t34-416t33-402t32-342t31-917t30-502t29-909t28-1248t27-662t26-1698t25-1332t24-1224t23-2241t22-1192t21-1584t20-1704t19-640t18-1024t17-528t16-128t15-256t14)*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+(t20+2t15+2t13+7t12+2t10-2t9+2t8+16t7+8t5+32t4)*x4+(t32-t30+2t29-6t27+t26-3t24-2t23-16t22+18t21-14t20-74t19+7t18-34t17-64t16-18t15-80t14+72t13-96t12-252t11+16t10-130t9-256t8)*x3+(-2t2-2)*x2y+(-t42-10t39-2t37-32t36-2t35-26t34-38t33-20t32-140t31-17t30-76t29-331t28-22t27-178t26-342t25-120t24-438t23-142t22-270t21-778t20+8t19-193t18-672t17+87t16+42t15-204t14+96t13+65t12+8t11+32t10)*x2+(2t17+2t15+8t14+10t12+4t10+16t9+2t8+24t7+32t6+8t5+32t4)*xy+y2+(-t46-2t44-10t43-t42-22t41-33t40-16t39-94t38-46t37-110t36-228t35-88t34-416t33-402t32-342t31-917t30-502t29-909t28-1248t27-662t26-1698t25-1332t24-1224t23-2241t22-1192t21-1584t20-1704t19-640t18-1024t17-528t16-128t15-256t14)*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+(t42+t40+3t37+7t35+12t34+4t33+15t32+12t30+25t29+15t28+61t27+63t26+42t25+62t24+28t23+56t22+36t21+97t20+120t19+80t18+118t17+42t16+76t15+8t14+72t13+64t12+4t11+64t10-4t9)*x4+(-4t42-4t40-12t37-28t35-48t34-16t33-60t32-48t30-100t29-60t28-244t27-248t26-168t25-236t24-112t23-212t22-146t21-384t20-478t19-332t18-454t17-206t16-290t15-76t14-320t13-280t12-96t11-296t10-64t9-98t8-40t7-96t6-8t5-32t4)*x3y+(6t42+6t40+18t37+42t35+72t34+24t33+90t32+72t30+150t29+90t28+366t27+368t26+252t25+342t24+168t23+306t22+221t21+572t20+715t19+510t18+663t17+347t16+421t15+158t14+512t13+444t12+224t11+488t10+176t9+265t8+100t7+280t6+20t5+120t4+20t2+4)*x2y2+(-4t42-4t40-12t37-28t35-48t34-16t33-60t32-48t30-100t29-60t28-244t27-244t26-168t25-224t24-112t23-200t22-148t21-380t20-476t19-344t18-436t17-244t16-276t15-120t14-352t13-304t12-176t11-340t10-144t9-216t8-80t7-232t6-16t5-104t4-20t2-4)*xy3+(t42+t40+3t37+7t35+12t34+4t33+15t32+12t30+25t29+15t28+61t27+61t26+42t25+56t24+28t23+50t22+37t21+95t20+119t19+86t18+109t17+61t16+69t15+30t14+88t13+76t12+44t11+85t10+36t9+54t8+20t7+58t6+4t5+26t4+5t2+1)*y4+(t71+t69+7t68-t67+6t66+12t65-7t64+22t63+t62-t61+108t60-58t59+65t58+134t57-286t56+143t55-319t54-436t53+305t52-1251t51-62t50-570t49-3119t48-1009t47-4962t46-4922t45-4531t44-8891t43-5107t42-10800t41-12823t40-13152t39-23043t38-19325t37-25894t36-23171t35-19721t34-27749t33-16817t32-28124t31-29534t30-22096t29-30326t28-17651t27-15691t26-11340t25-5844t24-11534t23-3457t22-9140t21-6481t20-1048t19-3900t18+508t17-16t16)*x3+(-3t71-3t69-21t68+3t67-18t66-36t65+21t64-66t63-3t62+3t61-324t60+174t59-203t58-402t57+826t56-499t55+909t54+1012t53-1111t52+3237t51-676t50+1152t49+7299t48+1651t47+11842t46+10202t45+9359t44+17787t43+7379t42+21408t41+23661t40+27120t39+46711t38+38219t37+49664t36+36179t35+28555t34+41333t33+17911t32+44870t31+49064t30+31392t29+42242t28+16053t27+1265t26-5320t25-12586t24-202t23-7565t22-148t21+3023t20-16004t19-5128t18-10504t17-11204t16-2280t15-4016t14-320t13-772t12-32t11-128t10)*x2y+(3t71+3t69+21t68-3t67+18t66+36t65-21t64+66t63+3t62-3t61+324t60-174t59+207t58+402t57-810t56+534t55-885t54-864t53+1209t52-2979t51+1107t50-873t49-6270t48-963t47-10320t46-7920t45-7242t44-13344t43-3408t42-15912t41-16257t40-20952t39-35502t38-28341t37-35655t36-19512t35-13251t34-20376t33-1641t32-25119t31-29295t30-13944t29-17874t28+2397t27+21639t26+24990t25+27645t24+17604t23+16533t22+13932t21+5187t20+25578t19+13542t18+14994t17+16830t16+3420t15+6024t14+480t13+1158t12+48t11+192t10)*xy2+(-t71-t69-7t68+t67-6t66-12t65+7t64-22t63-t62+t61-108t60+58t59-69t58-134t57+270t56-178t55+295t54+288t53-403t52+993t51-369t50+291t49+2090t48+321t47+3440t46+2640t45+2414t44+4448t43+1136t42+5304t41+5419t40+6984t39+11834t38+9447t37+11885t36+6504t35+4417t34+6792t33+547t32+8373t31+9765t30+4648t29+5958t28-799t27-7213t26-8330t25-9215t24-5868t23-5511t22-4644t21-1729t20-8526t19-4514t18-4998t17-5610t16-1140t15-2008t14-160t13-386t12-16t11-64t10)*y3+(-t98-3t96-19t95-3t94-61t93-145t92-75t91-521t90-604t89-817t88-2523t87-1940t86-5163t85-8231t84-7752t83-21237t82-21786t81-33286t80-61366t79-57291t78-112131t77-138981t76-162227t75-287543t74-291752t73-434075t72-591047t71-622336t70-955504t69-1047496t68-1281561t67-1716917t66-1776078t65-2361310t64-2668923t63-2931744t62-3657485t61-3731222t60-4351535t59-4750097t58-4873089t57-5603112t56-5502402t55-5906901t54-6105204t53-5726464t52-6057971t51-5439173t50-5204703t49-5063091t48-4160646t47-4176547t46-3452152t45-2734119t44-2549615t43-1525457t42-1145610t41-684634t40-88614t39-43941t38+402490t37+354535t36+481592t35+730705t34+458614t33+744017t32+617544t31+433917t30+522826t29+255409t28+208660t27+137923t26+45072t25+40193t24+6672t23+7296t22+512t21+1024t20)*x2+(2t98+6t96+38t95+6t94+122t93+290t92+150t91+1044t90+1208t89+1644t88+5080t87+3900t86+10504t85+16702t84+15896t83+43750t82+44710t81+69818t80+127782t79+120408t78+239576t77+293292t76+354138t75+623858t74+636224t73+979440t72+1307672t71+1431684t70+2209864t69+2412748t68+3111052t67+4091520t66+4396744t65+6008688t64+6725168t63+7909676t62+9793424t61+10366632t60+12737090t59+13841454t58+15428400t57+17991452t56+18562556t55+21539888t54+22481318t53+23710894t52+25924230t51+25394764t50+27638970t49+27655238t48+27988230t47+29505508t46+28028786t45+28831996t44+27216364t43+25912168t42+25008304t41+22436118t40+22269590t39+19809364t38+18624194t37+17313754t36+14347408t35+13133994t34+10442832t33+8765530t32+7298932t31+5754810t30+5185524t29+3653920t28+3253968t27+2153466t26+1364656t25+1024510t24+338528t23+310560t22+54656t21+57344t20+6144t19+8192t18)*xy+(-t98-3t96-19t95-3t94-61t93-145t92-75t91-522t90-604t89-822t88-2540t87-1950t86-5252t85-8351t84-7948t83-21875t82-22355t81-34909t80-63891t79-60204t78-119788t77-146646t76-177069t75-311929t74-318112t73-489720t72-653836t71-715842t70-1104932t69-1206374t68-1555526t67-2045760t66-2198372t65-3004344t64-3362584t63-3954838t62-4896712t61-5183316t60-6368545t59-6920727t58-7714200t57-8995726t56-9281278t55-10769944t54-11240659t53-11855447t52-12962115t51-12697382t50-13819485t49-13827619t48-13994115t47-14752754t46-14014393t45-14415998t44-13608182t43-12956084t42-12504152t41-11218059t40-11134795t39-9904682t38-9312097t37-8656877t36-7173704t35-6566997t34-5221416t33-4382765t32-3649466t31-2877405t30-2592762t29-1826960t28-1626984t27-1076733t26-682328t25-512255t24-169264t23-155280t22-27328t21-28672t20-3072t19-4096t18)*y2+(-t119-6t117-23t116-15t115-143t114-241t113-382t112-1475t111-1723t110-4295t109-8968t108-10941t107-28630t106-39429t105-64174t104-130746t103-156826t102-310326t101-463369t100-628890t99-1169550t98-1454685t97-2333450t96-3547511t95-4446150t94-7350213t93-9421312t92-13131965t91-19511206t90-23747672t89-35045288t88-45359175t87-57759296t86-81402111t85-97610272t84-130731994t83-167019396t82-200917716t81-265854241t80-313952949t79-389382522t78-481421415t77-555005593t76-687903875t75-791984249t74-924020951t73-1094054234t72-1214514302t71-1417572297t70-1574970640t69-1740697646t68-1959677297t67-2083079335t66-2296106769t65-2441416842t64-2560713351t63-2739699200t62-2774411977t61-2889341715t60-2918985217t59-2885745588t58-2917666548t57-2788064352t56-2732191808t55-2610106147t54-2422537541t53-2309665538t52-2067019498t51-1892253886t50-1685191429t49-1442960061t48-1279047126t47-1046398255t46-885858612t45-725652940t44-561144012t43-461423713t42-331820896t41-250491872t40-179601740t39-112273664t38-79788160t37-42918448t36-25408512t35-13931264t34-5408832t33-3359744t32-793600t31-552960t30-65536t29-65536t28)*x+(t119+6t117+23t116+15t115+143t114+241t113+382t112+1475t111+1723t110+4295t109+8968t108+10941t107+28630t106+39429t105+64174t104+130746t103+156826t102+310326t101+463369t100+628890t99+1169550t98+1454685t97+2333450t96+3547511t95+4446150t94+7350213t93+9421312t92+13131965t91+19511206t90+23747672t89+35045288t88+45359175t87+57759296t86+81402111t85+97610272t84+130731994t83+167019396t82+200917716t81+265854241t80+313952949t79+389382522t78+481421415t77+555005593t76+687903875t75+791984249t74+924020951t73+1094054234t72+1214514302t71+1417572297t70+1574970640t69+1740697646t68+1959677297t67+2083079335t66+2296106769t65+2441416842t64+2560713351t63+2739699200t62+2774411977t61+2889341715t60+2918985217t59+2885745588t58+2917666548t57+2788064352t56+2732191808t55+2610106147t54+2422537541t53+2309665538t52+2067019498t51+1892253886t50+1685191429t49+1442960061t48+1279047126t47+1046398255t46+885858612t45+725652940t44+561144012t43+461423713t42+331820896t41+250491872t40+179601740t39+112273664t38+79788160t37+42918448t36+25408512t35+13931264t34+5408832t33+3359744t32+793600t31+552960t30+65536t29+65536t28)*y

drawTropicalCurve(newfnox,"max");