/////////////////////////////////// // Modifications for Theta Graph // /////////////////////////////////// // The following contains an example for Cell4 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][0].Hrepresentation() // (An equation (0, 0, 1, -1) x + 0 == 0, // An inequality (1, -1, 0, 0) x + 0 >= 0, // An inequality (-1, 2, -1, 0) x + 0 >= 0) ///////////////////////////// // Case 4: [0, -1, -3, -3] // ///////////////////////////// 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 = (3*t+t^7); poly B34 = (5*t^3+t^9); poly B2 = (17*t^3+t^6); 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+(t18+2t16+2t14+11t12+16t10+34t9+12t8+314t6+29t4+16t2-1)*x4+(-t32-3t30-3t28-34t27-18t26-68t25-327t24-68t23-605t22-340t21-680t20-544t19-3052t18-408t17-4691t16-850t15-3686t14-986t13-7494t12-544t11-8424t10+34t9-4757t8+122t6-15t4+18t2)*x3+(t44+2t42+34t41+t40+68t39+305t38+34t37+605t36+544t35+296t34+952t33+4708t32+374t31+8209t30+3060t29+3156t28+4658t27+26128t26+1292t25+39717t24+6868t23+10681t22+9180t21+58075t20+1462t19+77705t18+4522t17+11584t16+5678t15+37423t14+510t13+47498t12-612t11+3507t10-6048t8-432t6-81t4)*x2+y2+(t48+4t46+34t45+6t44+136t43+309t42+204t41+1217t40+680t39+1818t38+2074t37+5926t36+2856t35+17973t34+4964t33+24712t32+11696t31+42686t30+14824t29+100274t28+16728t27+127000t26+29172t25+142953t24+33864t23+248790t22+26010t21+288690t20+28152t19+221517t18+28764t17+239373t16+14688t15+244494t14+2754t13+124848t12+23409t10)*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+(t18+2t16+2t14+11t12+16t10+34t9+12t8+314t6+30t4+18t2)*x4+(-t32-3t30-3t28-34t27-18t26-68t25-327t24-68t23-605t22-340t21-682t20-544t19-3058t18-408t17-4697t16-850t15-3704t14-986t13-7538t12-544t11-8464t10+34t9-4799t8+44t6-81t4)*x3+(-2t2-2)*x2y+(t44+2t42+34t41+t40+68t39+305t38+34t37+606t36+544t35+300t34+952t33+4714t32+374t31+8229t30+3060t29+3217t28+4658t27+26212t26+1292t25+39863t24+6868t23+11025t22+9180t21+58511t20+1462t19+78197t18+4522t17+12442t16+5678t15+38419t14+510t13+48263t12-612t11+4335t10-5202t8)*x2+(2t18+4t16+2t14+16t12+28t10+12t8+30t6+48t4+18t2)*xy+y2+(t48+4t46+34t45+6t44+136t43+309t42+204t41+1217t40+680t39+1818t38+2074t37+5926t36+2856t35+17973t34+4964t33+24712t32+11696t31+42686t30+14824t29+100274t28+16728t27+127000t26+29172t25+142953t24+33864t23+248790t22+26010t21+288690t20+28152t19+221517t18+28764t17+239373t16+14688t15+244494t14+2754t13+124848t12+23409t10)*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+(t38+4t36+6t34+23t32+73t30+100t28-34t27+198t26-68t25+216t24+34t23+56t22-204t21+2227t20-476t19+981t18+19448t17-2234t16+19210t15+83189t14-612t13+81586t12-102t11-3252t10+68t9-42t8+628t6)*x4+(-4t38-16t36-24t34-92t32-292t30-400t28+136t27-792t26+272t25-860t24-136t23-214t22+816t21-8904t20+1904t19-3888t18-77792t17+9044t16-76704t15-332686t14+2856t13-325128t12+816t11+16694t10-136t9+3816t8-1400t6-84t4-18t2)*x3y+(6t38+24t36+36t34+138t32+438t30+600t28-204t27+1188t26-408t25+1286t24+204t23+311t22-1224t21+13352t20-2856t19+5796t18+116688t17-13674t16+114920t15+498959t14-4692t13+486476t12-1632t11-28723t10+68t9-9352t8+1028t6+250t4+65t2+4)*x2y2+(-4t38-16t36-24t34-92t32-292t30-400t28+136t27-792t26+272t25-856t24-136t23-204t22+816t21-8900t20+1904t19-3852t18-77792t17+9152t16-76568t15-332616t14+3264t13-323912t12+1224t11+20376t10+7444t8-328t6-208t4-56t2-4)*xy3+(t38+4t36+6t34+23t32+73t30+100t28-34t27+198t26-68t25+214t24+34t23+51t22-204t21+2225t20-476t19+963t18+19448t17-2288t16+19142t15+83154t14-816t13+80978t12-306t11-5094t10-1861t8+82t6+52t4+14t2+1)*y4+(-t70-6t68-16t66-34t65-58t64-204t63-500t62-442t61-2212t60-1258t59-3757t58-5168t57-8077t56+8602t55-38572t54+78234t53+11272t52+139162t51+333079t50+453254t49+505547t48+1732266t47+1705526t46+3099678t45+7306482t44+5231648t43+12773616t42+13038898t41+20792226t40+21318680t39+53975410t38+25943156t37+88174243t36+43522210t35+104159663t34+62737072t33+177146520t32+58374770t31+257615480t30+58566326t29+234419739t28+68438702t27+233181017t26+48057742t25+277279763t24+14067738t23+192026113t22+1746138t21+50320995t20+1145664t19+3124616t18+267036t17+3839967t16+1156t15+854016t14-2754t13-106145t12-25434t10)*x3+(3t70+18t68+48t66+102t65+174t64+612t63+1500t62+1326t61+6636t60+3774t59+11263t58+15504t57+24183t56-25806t55+115602t54-234702t53-34154t52-417282t51-1000513t50-1358742t49-1517747t48-5194758t47-5113436t46-9293322t45-21916630t44-15675972t43-38301930t42-39081402t41-62262636t40-63897152t39-161717680t38-77694760t37-264188409t36-130337266t35-311565691t34-187921332t33-529831742t32-174675442t31-770914606t30-175028294t29-700103945t28-204617474t27-694593941t26-143494790t25-826840775t24-41429374t23-571331287t22-4559298t21-145272367t20-3093320t19-4377782t18-706860t17-9271421t16+15232t15-2181312t14+12954t13+324835t12+612t11+89634t10+3852t8)*x2y+(-3t70-18t68-48t66-102t65-174t64-612t63-1500t62-1326t61-6636t60-3774t59-11259t58-15504t57-24159t56+25806t55-115545t54+234702t53+34323t52+417180t51+1001151t50+1358232t49+1518300t48+5193738t47+5111865t46+9290466t45+21915222t44+15666486t43+38292471t42+39063756t41+62205615t40+63867708t39+161613405t38+77627406t37+264021249t36+130222584t35+311109042t34+187776390t33+529027833t32+174451008t31+769948689t30+174692952t29+698526309t28+204268158t27+692119386t26+143155572t25+824341518t24+41042454t23+568957761t22+4219740t21+142427058t20+2921484t19+1879749t18+659736t17+8147181t16-24582t15+1990944t14-15300t13-328035t12-918t11-96300t10-5778t8)*xy2+(t70+6t68+16t66+34t65+58t64+204t63+500t62+442t61+2212t60+1258t59+3753t58+5168t57+8053t56-8602t55+38515t54-78234t53-11441t52-139060t51-333717t50-452744t49-506100t48-1731246t47-1703955t46-3096822t45-7305074t44-5222162t43-12764157t42-13021252t41-20735205t40-21289236t39-53871135t38-25875802t37-88007083t36-43407528t35-103703014t34-62592130t33-176342611t32-58150336t31-256649563t30-58230984t29-232842103t28-68089386t27-230706462t26-47718524t25-274780506t24-13680818t23-189652587t22-1406580t21-47475686t20-973828t19-626583t18-219912t17-2715727t16+8194t15-663648t14+5100t13+109345t12+306t11+32100t10+1926t8)*y3+(t100+7t98+34t97+20t96+238t95+360t94+646t93+2329t92+2210t91+5135t90+9894t89+13771t88+5712t87+72759t86-59806t85+83931t84-107270t83-225079t82-613496t81-150239t80-3331728t79-1842783t78-7400576t77-14027532t76-17221782t75-29659250t74-55965292t73-66447041t72-116331374t71-234721404t70-208242452t69-484076204t68-498456048t67-837981132t66-944864352t65-2071463365t64-1429009834t63-3939341513t62-2672898100t61-5815743020t60-4560644910t59-11011615953t58-6006241576t57-18929366175t56-8921829024t55-24508068422t54-13466931344t53-36463766420t52-15568582694t51-55560210556t50-18153340962t49-63498674358t48-23456431292t47-73618039485t46-23632952050t45-96117202532t44-20600759970t43-96250396439t42-21051798190t41-82830794394t40-17925385310t39-85525531842t38-10019163622t37-72966515627t36-5762421388t35-39849380768t34-3609190576t33-22985449510t32-66211736t31-14987781391t30+1697048120t29-380073641t28+950898740t27+6820567675t26+142725540t25+3482563615t24-19109700t23+217171065t22-5324400t21-218809125t20-275400t19-45257400t18-2340900t16)*x2+(-2t100-14t98-68t97-40t96-476t95-720t94-1292t93-4656t92-4420t91-10252t90-19788t89-27472t88-11424t87-145282t86+119544t85-166926t84+213996t83+452548t82+1225088t81+303538t80+6657336t79+3690840t78+14778304t77+28060118t76+34377876t75+59252472t74+111773572t73+132624360t72+232252776t71+468764662t70+415493396t69+965987658t68+994894672t67+1670078194t66+1885560984t65+4130791056t64+2849433308t63+7852365470t62+5330369244t61+11573979328t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drawTropicalCurve(newfnox,"max");