/////////////////////////////////// // Modifications for Theta Graph // /////////////////////////////////// // The following contains an example for Cell5 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][1].Hrepresentation() // (An equation (1, -2, 1, 0) x + 0 == 0, // An inequality (1, 0, -1, 0) x + 0 >= 0, // An inequality (0, 0, 1, -1) x + 0 >= 0) // We want cancellation, namely in(b4)^2 = in(b5*b34) //////////////////////////////////////////////////////// // Case 5 (with cancellation): 2w3 = w34+w5, 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 = (2*t+t^7); poly B34 = (4*t^2-t^5); poly B2 = (17*t^3+13*t^4); 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+(2t14-2t12+t10+8t9+177t8+434t7+285t6+15t4+16t3+6t2-1)*x4+(-t28+2t26-t24-8t23-346t22-876t21-228t20+884t19+391t18-1842t17-5199t16-4464t15+1923t14+4088t13-1439t12-9422t11-12440t10-7252t9-1533t8+450t7+261t6-32t5+16t4+16t3+8t2)*x3+(169t36+442t35-49t34-884t33-410t32+1794t31+5177t30+4496t29-3250t28-7624t27-434t26+16060t25+30131t24+18196t23-11677t22-20806t21-95t20+39016t19+59403t18+29790t17-10569t16-22496t15-8196t14+15312t13+19416t12-560t11-13112t10-8192t9-2440t8-128t7-96t6-64t5-16t4)*x2+y2+(169t40+442t39+289t38-338t36+468t35+4310t34+7200t33+6693t32+3170t31-1415t30+6216t29+27624t28+45360t27+50752t26+35240t25+13080t24+25640t23+65340t22+109184t21+131940t20+102920t19+58484t18+42944t17+54272t16+87456t15+109792t14+90240t13+58736t12+25568t11+4624t10)*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+(2t14-2t12+t10+8t9+177t8+434t7+285t6+16t4+16t3+8t2)*x4+(-t28+2t26-t24-8t23-346t22-876t21-228t20+884t19+389t18-1842t17-5201t16-4464t15+1925t14+4080t13-1445t12-9438t11-12452t10-7260t9-1533t8+434t7+257t6-64t5)*x3+(-2t2-2)*x2y+(169t36+442t35-49t34-884t33-409t32+1794t31+5177t30+4496t29-3252t28-7616t27-426t26+16068t25+30136t24+18188t23-11673t22-20766t21-43t20+39080t19+59447t18+29774t17-10525t16-22432t15-8052t14+15472t13+19532t12-528t11-13048t10-8160t9-2312t8)*x2+(2t16-2t12+8t11+8t10+8t9+4t8-4t6+16t5+8t4+16t3+8t2)*xy+y2+(169t40+442t39+289t38-338t36+468t35+4310t34+7200t33+6693t32+3170t31-1415t30+6216t29+27624t28+45360t27+50752t26+35240t25+13080t24+25640t23+65340t22+109184t21+131940t20+102920t19+58484t18+42944t17+54272t16+87456t15+109792t14+90240t13+58736t12+25568t11+4624t10)*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+(3t26-169t24-466t23-291t22+12t21+230t20-214t19+26316t18+145684t17+318293t16+403000t15+376971t14+255884t13+81401t12-3180t11-2618t10-1188t9+1460t8+1252t7+866t6+32t4)*x4+(-12t26+676t24+1864t23+1164t22-48t21-922t20+856t19-105268t18-582736t17-1273168t16-1612008t15-1507200t14-1021824t13-322426t12+17904t11+16024t10+9936t9-1508t8-3320t7-2136t6-48t5-88t4-16t3-8t2)*x3y+(18t26-1014t24-2796t23-1746t22+72t21+1385t20-1284t19+157906t18+874104t17+1909748t16+2418020t15+2260116t14+1531024t13+480461t12-32040t11-29584t10-20088t9-2050t8+3292t7+1916t6+120t5+132t4+40t3+40t2+4)*x2y2+(-12t26+676t24+1864t23+1164t22-48t21-924t20+856t19-105272t18-582736t17-1273164t16-1612016t15-1506516t14-1020112t13-319248t12+23088t11+21572t10+15120t9+2804t8-1632t7-848t6-96t5-88t4-32t3-36t2-4)*xy3+(3t26-169t24-466t23-291t22+12t21+231t20-214t19+26318t18+145684t17+318291t16+403004t15+376629t14+255028t13+79812t12-5772t11-5393t10-3780t9-701t8+408t7+212t6+24t5+22t4+8t3+9t2+1)*y4+(-t56+t54+8t53+2t52-20t51-537t50-1342t49-750t48+4192t47+11461t46+4738t45+63476t44+399212t43+949505t42+630772t41-2330823t40-6835426t39-6810474t38+3349086t37+18206666t36+17750838t35-11181607t34-50012860t33-52479380t32+13844036t31+109698834t30+149988386t29+87870984t28-34842664t27-88247179t26+15122348t25+226793866t24+423422896t23+483771520t22+392426060t21+238558561t20+105540652t19+32746564t18+10234520t17+5384766t16+2847956t15+1245824t14+371236t13+120657t12-19744t11+4112t10-1024t9)*x3+(3t56-3t54-24t53-6t52+60t51+1611t50+4026t49+2250t48-12576t47-34391t46-14214t45-190444t44-1197572t43-2848507t42-1892252t41+6993323t40+20508674t39+20436302t38-10047286t37-54632502t36-53281426t35+33513801t34+150049816t33+157497344t32-41446860t31-329099728t30-450149870t29-263891732t28+104315140t27+264948311t26-44677464t25-679414044t24-1269335852t23-1450742322t22-1176878736t21-714963779t20-315146312t19-95937702t18-27777232t17-13496102t16-6358756t15-2430284t14-474564t13-88679t12+97072t11+6888t10-1376t9-2040t8-768t7-384t6)*x2y+(-3t56+3t54+24t53+6t52-60t51-1611t50-4026t49-2250t48+12576t47+34395t46+14214t45+190452t44+1197540t43+2848503t42+1892220t41-6993750t40-20509872t39-20438742t38+10047300t37+54638754t36+53295882t35-33498291t34-150055434t33-157526946t32+41404236t31+329101341t30+450242226t29+264031122t28-104208714t27-265051698t26+44332674t25+678930267t24+1268869434t23+1450456203t22+1176679014t21+714607827t20+314408490t19+94786707t18+26314068t17+12167004t16+5266200t15+1776690t14+154992t13-47967t12-115992t11-16500t10+3600t9+3060t8+1152t7+576t6)*xy2+(t56-t54-8t53-2t52+20t51+537t50+1342t49+750t48-4192t47-11465t46-4738t45-63484t44-399180t43-949501t42-630740t41+2331250t40+6836624t39+6812914t38-3349100t37-18212918t36-17765294t35+11166097t34+50018478t33+52508982t32-13801412t31-109700447t30-150080742t29-88010374t28+34736238t27+88350566t26-14777558t25-226310089t24-422956478t23-483485401t22-392226338t21-238202609t20-104802830t19-31595569t18-8771356t17-4055668t16-1755400t15-592230t14-51664t13+15989t12+38664t11+5500t10-1200t9-1020t8-384t7-192t6)*y3+(169t80+442t79+120t78-1794t77-4332t76+418t75-12905t74-122934t73-297504t72-119710t71+1064686t70+2476810t69+712262t68-6771394t67-14071287t66-3714656t65+33717284t64+67733942t63+32857715t62-99515986t61-218812555t60-109480602t59+297923483t58+689055626t57+475991637t56-557934024t55-1641980746t54-1440847718t53+574475212t52+2926423466t51+2774313081t50-1136524708t49-6399582308t48-8292177324t47-4040050147t46+3095386112t45+5815445335t44-345537108t43-12337663702t42-20833812884t41-18633770729t40-8128245322t39+1339122814t38+1566336840t37-7284584720t36-16422016772t35-17780211986t34-10499735508t33+109793399t32+7273257900t31+9275692126t30+9245186984t29+10160747782t28+12627100040t27+14622283276t26+13919088696t25+10660247404t24+6438444704t23+2943877856t22+940511648t21+138732928t20-49042304t19-46755648t18-23766528t17-6718464t16-1636352t15-295936t14)*x2+(-338t80-884t79-240t78+3588t77+8666t76-836t75+25812t74+245852t73+595000t72+239436t71-2129654t70-4954376t69-1425942t68+13543468t67+28148636t66+7437344t65-67435776t64-135501364t63-65762976t62+199038404t61+437762386t60+219218168t59-595752728t58-1378450968t57-952698112t56+1115497392t55+3285034478t54+2884127224t53-1146963010t52-5854059604t51-5554028912t50+2267208208t49+12799147124t48+16592574320t47+8090774344t46-6191678088t45-11653242176t44+653727132t43+24641223580t42+41655367460t41+37271678716t40+16244556132t39-2740509220t38-3255612752t37+14412779072t36+32701906760t35+35452948292t34+20903511832t33-352692014t32-14761690376t31-18837217340t30-18799792480t29-20600516740t28-25477770000t27-29430183576t26-28024689456t25-21526314392t24-13098907264t23-6096875648t22-2046409984t21-389476192t20+37327360t19+66550784t18+38534656t17+11881600t16+3381248t15+887296t14+166912t13+47104t12+12288t11+2048t10)*xy+(169t80+442t79+120t78-1794t77-4333t76+418t75-12906t74-122926t73-297500t72-119718t71+1064827t70+2477188t69+712971t68-6771734t67-14074318t66-3718672t65+33717888t64+67750682t63+32881488t62-99519202t61-218881193t60-109609084t59+297876364t58+68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drawTropicalCurve(newfnox,"max");