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

// The following contains an example for Cell3 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][3].Hrepresentation()
// (An inequality (0, 0, 1, -1) x + 0 >= 0,
//  An inequality (1, -2, 1, 0) x + 0 >= 0,
//  An inequality (0, 1, -1, 0) x + 0 >= 0)

/////////////////////////////
// Case 3: [0, -2, -3, -4] //
/////////////////////////////

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 = (t^2+t^6);
poly B34 = (7*t^3+11*t^9);
poly B2 = (5*t^4-17*t^5);

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+(121t18+22t15+156t12+22t11+289t10-156t9+29t8+49t6+14t5+t4-2t2-1)*x4+(-121t30-34969t28+20548t27-3267t26-6358t25+3585t24-616t23-45084t22+20148t21-230t20-4640t19+1296t18+332t17-14265t16+4306t15+1080t14-164t13+709t12-290t11+347t10-156t9+128t8+28t7+53t6+14t5+2t4)*x3+(34969t40-20570t39+3025t38+6358t37+66198t36-40590t35+50724t34-7276t33-7587t32+5674t31+17487t30+1122t29-27177t28+12674t27-977t26-7234t25-58715t24+31046t23-48508t22+11482t21+1718t20-4108t19-28054t18+8940t17-13492t16+4158t15+322t14-66t13-153t12-28t11-51t10-14t9-t8)*x2+y2+(34969t44-20570t43+72963t42-34782t41+107217t40-48444t39+186266t38-82098t37+195512t36-70016t35+196906t34-70640t33+219991t32-76648t31+152620t30-42492t29+119973t28-38098t27+93955t26-27366t25+35775t24-7516t23+24122t22-8018t21+12554t20-3924t19-816t18+180t17+25t16)*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+(121t18+22t15+156t12+22t11+289t10-156t9+29t8+49t6+14t5+2t4)*x4+(-121t30-34969t28+20548t27-3267t26-6358t25+3585t24-616t23-45084t22+20148t21-230t20-4662t19+1296t18+288t17-14267t16+4262t15+1076t14-222t13+703t12-340t11+339t10-184t9+122t8+49t6)*x3+(-2t2-2)*x2y+(34969t40-20570t39+3025t38+6358t37+66198t36-40590t35+50845t34-7276t33-7345t32+5696t31+17850t30+1166t29-26538t28+12762t27-304t26-7088t25-58006t24+31206t23-47714t22+11670t21+2288t20-3936t19-27587t18+9068t17-13132t16+4264t15+477t14-10t13-50t12)*x2+(22t17+22t15+2t14+22t13+2t12+36t11+4t10+14t9+4t8+14t7+2t6+14t5+2t4)*xy+y2+(34969t44-20570t43+72963t42-34782t41+107217t40-48444t39+186266t38-82098t37+195512t36-70016t35+196906t34-70640t33+219991t32-76648t31+152620t30-42492t29+119973t28-38098t27+93955t26-27366t25+35775t24-7516t23+24122t22-8018t21+12554t20-3924t19-816t18+180t17+25t16)*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+(14641t38+14641t36+3993t35+3993t33+37631t32+3993t31+37631t30+11616t29+726t28+4444t27+38632t26+4169t25+38222t24+9190t23+87090t22-98716t21+146217t20-104042t19+60788t18-4478t17+3734t16-1339t15+6279t14+4t13+4813t12+9t11+1169t10-340t9+344t8+98t6)*x4+(-58564t38-58564t36-15972t35-15972t33-150524t32-15972t31-150524t30-46464t29-2904t28-17776t27-154528t26-16676t25-152404t24-36760t23-346908t22+394842t21-583416t20+416102t19-242054t18+17824t17-11938t16+4574t15-19710t14-2164t13-14686t12-2154t11-2646t10+624t9-698t8-42t7-202t6-14t5-2t4)*x3y+(87846t38+87846t36+23958t35+23958t33+225786t32+23958t31+225786t30+69696t29+4356t28+26664t27+231792t26+25014t25+228122t24+55140t23+518910t22-592241t21+873672t20-624087t19+361983t18-26648t17+14909t16-6079t15+24159t14+5394t13+17463t12+5349t11+1943t10-200t9+389t8+105t7+153t6+35t5+45t4+20t2+4)*x2y2+(-58564t38-58564t36-15972t35-15972t33-150524t32-15972t31-150524t30-46464t29-2904t28-17776t27-154528t26-16676t25-151920t24-36760t23-345456t22+394820t21-581964t20+416036t19-240956t18+17736t17-8940t16+3792t15-14304t14-4312t13-10120t12-4272t11-620t10-112t9-40t8-84t7-52t6-28t5-44t4-20t2-4)*xy3+(14641t38+14641t36+3993t35+3993t33+37631t32+3993t31+37631t30+11616t29+726t28+4444t27+38632t26+4169t25+37980t24+9190t23+86364t22-98705t21+145491t20-104009t19+60239t18-4434t17+2235t16-948t15+3576t14+1078t13+2530t12+1068t11+155t10+28t9+10t8+21t7+13t6+7t5+11t4+5t2+1)*y4+(-161051t67-46865841t65+27334747t64-97757957t63+37744498t62-92666882t61+1242054594t60-1672092708t59+3169411685t58-3009136779t57+2197383837t56-1120956705t55+3009153235t54-3258030149t53+7192738366t52-6556871541t51+4962441473t50-2437530843t49+2820454714t48-2330681078t47+6176524712t46-5531190489t45+4258256189t44-2164122914t43+1236342740t42-677187876t41+2358269608t40-2089858161t39+1615554122t38-855830423t37+221874451t36-37616658t35+326515301t34-292036875t33+222504624t32-116634361t31+3827053t30+13190059t29-3893614t28+1221820t27+459781t26+1553463t25+1988309t24-649880t23+1560053t22-706164t21+824589t20-334836t19+320796t18-94577t17+79441t16-17346t15+16807t14+2401t12)*x3+(483153t67+140597523t65-82004241t64+293273871t63-113233494t62+278000646t61-3726163782t60+5016278124t59-9508235055t58+9027410337t57-6592268639t56+3362870115t55-9027928217t54+9774069151t53-21579152122t52+19672837393t51-14890386687t50+7321832331t49-8469421048t48+7008492700t47-18542020438t46+16616705941t45-12791290171t44+6526214862t43-3731785394t42+2071649476t41-7100022716t40+6309401699t39-4870552932t38+2608084525t37-688588387t36+147771316t35-997777291t34+900429801t33-679643180t32+368587919t31-20261801t30-27363905t29+7106264t28+852998t27-2422171t26-2918725t25-6370203t24+2795976t23-4680913t22+1735308t21-1984757t20+400340t19-668148t18+22745t17-193349t16-15592t15-52413t14-12338t13-8917t12-2058t11-294t10)*x2y+(-483153t67-140597523t65+82004241t64-293273871t63+113233494t62-278000646t61+3726163782t60-5016278124t59+9508235055t58-9027410337t57+6592327203t56-3362870115t55+9028162473t54-9774058503t53+21579620634t52-19673948778t51+14891917821t50-7326452232t49+8473449501t48-7016717433t47+18548243589t46-16628273178t45+12799550973t44-6543137922t43+3743163981t42-2091692400t41+7112629662t40-6329315307t39+4882498215t38-2628381153t37+700070904t36-165231987t35+1006892985t34-912589389t33+685707834t32-377930337t31+24652122t30+21260769t29-4818975t28-3112227t27+2943585t26+2047893t25+6572841t24-3219144t23+4681290t22-1543716t21+1740252t20-98256t19+521028t18+107748t17+170862t16+49407t15+53409t14+18507t13+9774t12+3087t11+441t10)*xy2+(161051t67+46865841t65-27334747t64+97757957t63-37744498t62+92666882t61-1242054594t60+1672092708t59-3169411685t58+3009136779t57-2197442401t56+1120956705t55-3009387491t54+3258019501t53-7193206878t52+6557982926t51-4963972607t50+2442150744t49-2824483167t48+2338905811t47-6182747863t46+5542757726t45-4266516991t44+2181045974t43-1247721327t42+697230800t41-2370876554t40+2109771769t39-1627499405t38+876127051t37-233356968t36+55077329t35-335630995t34+304196463t33-228569278t32+125976779t31-8217374t30-7086923t29+1606325t28+1037409t27-981195t26-682631t25-2190947t24+1073048t23-1560430t22+514572t21-580084t20+32752t19-173676t18-35916t17-56954t16-16469t15-17803t14-6169t13-3258t12-1029t11-147t10)*y3+(511981129t94-301165370t93+1580232412t92-717321154t91+3607220298t90-14984668193t89+24442329604t88-50877255385t87+59690346771t86-102349080449t85+107927730031t84-213351991479t83+218311847545t82-348540625235t81+313600076812t80-439260223985t79+370608807305t78-561316005613t77+475915691108t76-632289772070t75+491557460449t74-573348517082t73+422379190563t72-490195244121t71+360699323516t70-335820850071t69+229556627381t68-133912216834t67+99644018272t66+43523784451t65-12856060790t64+195830417110t63-99761386219t62+276518873102t61-124654796539t60+328298330489t59-151001462852t58+306218869280t57-113952019148t56+219093619669t55-59811891233t54+186609723947t53-57614657639t52+124073191477t51-12295757318t50+37026622025t49+23506960429t48+20502652851t47+9501951077t46+9540505057t45+14148367708t44-13854499804t43+20201815004t42-12417441960t41+9564681538t40-5429956443t39+4357125954t38-4681019203t37+3544460687t36-3114393644t35+1509579333t34-1040116049t33+250413933t32-165819920t31+6083784t30-10540824t29-2125389t28-552965t27+120197t26+580699t25+75411t24-35280t23-4900t22)*x2+(-1023962258t94+602330740t93-3160464824t92+1434642308t91-7214440596t90+29969336386t89-48881116086t88+101754510770t87-119362977932t86+204699449306t85-215805856354t84+426617337520t83-436448940114t82+696627086650t81-626702535316t80+877334186234t79-740191134100t78+1120080934104t77-949920800686t76+1259725738524t75-979901721370t74+1138911783812t73-840107334726t72+969004460116t71-715123895274t70+656101886490t69-451250546128t68+248741095768t67-190468200134t66-109004356238t65+35062684482t64-415589248048t63+208771580422t62-576923279766t61+257513158712t60-679193316480t59+308892867982t58-632771530502t57+233190219610t56-454907361688t55+123063161552t54-386299097234t53+117258141042t52-257852327278t51+25480420098t50-80464862478t49-47005303498t48-45012466950t47-19359101300t46-21413454792t45-28804608938t44+26634788860t43-40936507548t42+24403827860t41-19521977624t40+10729388812t39-8980475898t38+9406320202t37-7262257980t36+6298061004t35-3103051148t34+2125910626t33-535837764t32+364292908t31-27570532t30+38397394t29+570528t28+6845300t27+300860t26+243628t25+314202t24+344470t23+110936t22+48020t21+15092t20+2058t19+98t18)*xy+(511981129t94-301165370t93+1580232412t92-717321154t91+3607220298t90-14984668193t89+24440558043t88-50877255385t87+59681488966t86-102349724653t85+107902928177t84-213308668760t83+218224470057t82-348313543325t81+313351267658t80-438667093117t79+370095567050t78-560040467052t77+474960400343t76-629862869262t75+489950860685t74-569455891906t73+420053667363t72-484502230058t71+357561947637t70-328050943245t69+225625273064t68-124370547884t67+95234100067t66+54502178119t65-17531342241t64+207794624024t63-104385790211t62+288461639883t61-128756579356t60+339596658240t59-154446433991t58+316385765251t57-116595109805t56+227453680844t55-61531580776t54+193149548617t53-58629070521t52+128926163639t51-12740210049t50+40232431239t49+23502651749t48+22506233475t47+9679550650t46+10706727396t45+14402304469t44-13317394430t43+20468253774t42-12201913930t41+9760988812t40-5364694406t39+4490237949t38-4703160101t37+3631128990t36-3149030502t35+1551525574t34-1062955313t33+267918882t32-182146454t31+13785266t30-19198697t29-285264t28-3422650t27-150430t26-121814t25-157101t24-172235t23-55468t22-24010t21-7546t20-1029t19-49t18)*y2+(5631792419t115-3312819070t114+34277933789t113-17317008775t112+114053097180t111-198638048686t110+481924341921t109-1090157601071t108+1717136552417t107-3473524820579t106+4465732521140t105-8634397439283t104+9945773996510t103-18587698738587t102+19722882392598t101-34659854624824t100+34268500797069t99-57929292925790t98+54361219968814t97-89277322245749t96+79963425939914t95-126341949712110t94+107999586468532t93-166205175778081t92+136588676288900t91-205865896015835t90+162744658398520t89-238724468295254t88+181043692209256t87-261041998357223t86+190666399946052t85-271114162806799t84+190275442748327t83-265363321871225t82+178052576874549t81-246204759656980t80+158270079934923t79-217452200835823t78+133083922995606t77-180720707209469t76+104262048347168t75-142190163333145t74+77263038515931t73-106057452907382t72+53362529697245t71-73441857587925t70+33253496997492t69-47550525711881t68+19021091303109t67-28615917640593t66+9317570195752t65-15119145844894t64+3109616095466t63-6929519393751t62+188703539644t61-2487738063583t60-1043061042136t59-156661750625t58-1386413819306t57+618138316098t56-1088347951150t55+668647277451t54-732220642452t53+551292684807t52-459719110442t51+344898122475t50-227869220452t49+170820068215t48-97851900157t47+80448984345t46-41812344487t45+32340098124t44-12866725645t43+8513577528t42-1913090893t41+1222205341t40-93031365t39+83753292t38-5581114t37+6354985t36-2837394t35-3180933t34-164052t33+150871t32+25970t31+1225t30)*x+(-5631792419t115+3312819070t114-34277933789t113+17317008775t112-114053097180t111+198638048686t110-481924341921t109+1090157601071t108-1717136552417t107+3473524820579t106-4465732521140t105+8634397439283t104-9945773996510t103+18587698738587t102-19722882392598t101+34659854624824t100-34268500797069t99+57929292925790t98-54361219968814t97+89277322245749t96-79963425939914t95+126341949712110t94-107999586468532t93+166205175778081t92-136588676288900t91+205865896015835t90-162744658398520t89+238724468295254t88-181043692209256t87+261041998357223t86-190666399946052t85+271114162806799t84-190275442748327t83+265363321871225t82-178052576874549t81+246204759656980t80-158270079934923t79+217452200835823t78-133083922995606t77+180720707209469t76-104262048347168t75+142190163333145t74-77263038515931t73+106057452907382t72-53362529697245t71+73441857587925t70-33253496997492t69+47550525711881t68-19021091303109t67+28615917640593t66-9317570195752t65+15119145844894t64-3109616095466t63+6929519393751t62-188703539644t61+2487738063583t60+1043061042136t59+156661750625t58+1386413819306t57-618138316098t56+108834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drawTropicalCurve(newfnox,"max");