/////////////////////////////////// // 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. We aim for a subdivision containing the edge x^2y^2---x^3. For this, we multiply our original vector by 3 (so we replace t by t^3) and take one term t^(3+e) with e = 1), since we need ht(x3)>2(w5+w4+w2)/3, and we want integer heights. //////////////////////////////////// // 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^6); poly B4 = (t^6+t^15); poly B34 = (t^9+t^10); poly B2 = (t^9); 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+(2t30+2t25+2t24+4t21+t20+2t19+2t16+2t15+t12-2t6-1)*x4+(-t60-2t55-2t54-4t51-t50-2t49+t48-6t46-6t45+2t43-2t42-2t41-4t40+2t39+t38-2t37+t36+2t34+2t33+4t31+7t30+8t27+2t26+6t25+3t24+4t22+8t21+t20+2t19+4t18+2t16+2t15+2t12)*x3+(-t78-2t73-3t72-4t69-t68-4t67-5t66-6t64-10t63-t62-6t61-10t60-2t59-10t58-16t57-2t56-10t55-10t54-2t53-16t52-18t51-t50-4t49-13t48-4t47-14t46-6t45+t44-8t43-14t42-2t41-2t40-4t39-t38-8t37-4t36-2t34-6t33-t32-2t31-t30-2t28-2t27-t24)*x2+y2+(-t90-2t85-4t84-4t81-t80-6t79-6t78-6t76-14t75-2t74-6t73-10t72-2t71-16t70-18t69-t68-8t67-19t66-4t65-14t64-14t63-t62-14t61-19t60-2t59-6t58-12t57-2t56-10t55-9t54-4t52-8t51-t50-2t49-3t48-2t46-2t45-t42)*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+(2t30+2t25+2t24+4t21+t20+2t19+2t16+2t15+2t12)*x4+(-t60-2t55-2t54-4t51-t50-2t49+t48-6t46-6t45+2t43-4t42-2t41-4t40+2t39+t38-4t37-5t36+2t34-2t33+t30-2t28-2t27+2t26+4t25-t24+t20+2t19)*x3+(-2t6-2)*x2y+(-t78-2t73-2t72-4t69-t68-2t67-t66-6t64-6t63-4t60-2t59-4t58-2t57-4t55+4t49+6t48+8t45+2t44+6t43+5t42+4t40+8t39+t38+2t37+5t36+2t34+2t33+2t30)*x2+(2t36+2t31+4t30+4t27+2t25+2t24+2t22+6t21+2t18+2t16+2t15+2t12)*xy+y2+(-t90-2t85-4t84-4t81-t80-6t79-6t78-6t76-14t75-2t74-6t73-10t72-2t71-16t70-18t69-t68-8t67-19t66-4t65-14t64-14t63-t62-14t61-19t60-2t59-6t58-12t57-2t56-10t55-9t54-4t52-8t51-t50-2t49-3t48-2t46-2t45-t42)*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+(3t56+6t55+4t54+3t51+12t50+16t49+8t48+6t47+13t46+15t45+15t44+14t43+9t42+15t41+23t40+16t39+11t38+14t37+9t36+9t35+10t34+4t33+9t32+18t31+4t30+6t26+12t25+2t20+4t19)*x4+(-12t56-24t55-16t54-12t51-48t50-64t49-34t48-24t47-52t46-60t45-60t44-58t43-44t42-60t41-92t40-68t39-40t38-54t37-48t36-36t35-42t34-30t33-24t32-54t31-26t30-6t28-18t27-12t26-26t25-8t24-6t22-10t21-4t20-8t19-6t18-2t16-2t15-2t12)*x3y+(18t56+36t55+24t54+18t51+72t50+96t49+53t48+36t47+78t46+90t45+90t44+89t43+74t42+90t41+138t40+106t39+56t38+79t37+84t36+54t35+65t34+59t33+24t32+63t31+53t30+15t28+45t27+6t26+17t25+40t24+15t22+25t21+2t20+4t19+55t18+5t16+5t15+45t12+20t6+4)*x2y2+(-12t56-24t55-16t54-12t51-48t50-64t49-36t48-24t47-52t46-60t45-60t44-60t43-52t42-60t41-92t40-72t39-36t38-52t37-60t36-36t35-44t34-44t33-12t32-36t31-40t30-12t28-36t27-4t25-36t24-12t22-20t21-52t18-4t16-4t15-44t12-20t6-4)*xy3+(3t56+6t55+4t54+3t51+12t50+16t49+9t48+6t47+13t46+15t45+15t44+15t43+13t42+15t41+23t40+18t39+9t38+13t37+15t36+9t35+11t34+11t33+3t32+9t31+10t30+3t28+9t27+t25+9t24+3t22+5t21+13t18+t16+t15+11t12+5t6+1)*y4+(-t122-2t121-t120-3t117-11t116-13t115-5t114-6t113-15t112-24t111-34t110-26t109-22t108-58t107-80t106-58t105-47t104-52t103-91t102-158t101-135t100-72t99-105t98-159t97-166t96-165t95-112t94-119t93-242t92-241t91-129t90-122t89-147t88-171t87-245t86-168t85-86t84-210t83-224t82-95t81-132t80-139t79-113t78-256t77-190t76-16t75-147t74-198t73-70t72-160t71-145t70-24t69-159t68-166t67-91t65-126t64+10t63-46t62-85t61+4t60-61t59-65t58+65t57+48t56-37t55-6t54-27t53-12t52+59t51+54t50-12t49-3t48-6t47+2t46+24t45+23t44-2t43+t40+4t39+4t38)*x3+(3t122+6t121+3t120+9t117+33t116+39t115+15t114+18t113+45t112+72t111+102t110+78t109+66t108+174t107+240t106+174t105+133t104+140t103+265t102+474t101+405t100+200t99+235t98+365t97+450t96+463t95+264t94+229t93+432t92+391t91+213t90+94t89+9t88+107t87+127t86-166t85-244t84-246t83-424t82-461t81-590t80-811t79-687t78-718t77-1048t76-974t75-1051t74-1338t73-1186t72-1114t71-1313t70-1154t69-1349t68-1722t67-1334t66-1083t65-1330t64-1280t63-1406t62-1551t61-1036t60-923t59-1321t58-1177t57-1028t56-1017t55-692t54-701t53-1006t52-747t51-542t50-564t49-419t48-372t47-498t46-310t45-209t44-258t43-196t42-120t41-147t40-76t39-52t38-80t37-58t36-18t35-20t34-8t33-6t32-12t31-8t30)*x2y+(-3t122-6t121-3t120-9t117-33t116-39t115-15t114-18t113-45t112-72t111-102t110-78t109-66t108-174t107-240t106-174t105-129t104-132t103-261t102-474t101-405t100-192t99-195t98-309t97-426t96-447t95-228t94-165t93-285t92-225t91-126t90+42t89+207t88+96t87+177t86+501t85+495t84+684t83+972t82+834t81+1083t80+1425t79+1200t78+1461t77+1857t76+1485t75+1797t74+2304t73+1884t72+1911t71+2187t70+1767t69+2262t68+2832t67+2001t66+1761t65+2184t64+1905t63+2178t62+2454t61+1548t60+1476t59+2079t58+1668t57+1470t56+1581t55+1047t54+1092t53+1527t52+1032t51+732t50+864t49+633t48+567t47+744t46+429t45+279t44+390t43+294t42+180t41+219t40+108t39+72t38+120t37+87t36+27t35+30t34+12t33+9t32+18t31+12t30)*xy2+(t122+2t121+t120+3t117+11t116+13t115+5t114+6t113+15t112+24t111+34t110+26t109+22t108+58t107+80t106+58t105+43t104+44t103+87t102+158t101+135t100+64t99+65t98+103t97+142t96+149t95+76t94+55t93+95t92+75t91+42t90-14t89-69t88-32t87-59t86-167t85-165t84-228t83-324t82-278t81-361t80-475t79-400t78-487t77-619t76-495t75-599t74-768t73-628t72-637t71-729t70-589t69-754t68-944t67-667t66-587t65-728t64-635t63-726t62-818t61-516t60-492t59-693t58-556t57-490t56-527t55-349t54-364t53-509t52-344t51-244t50-288t49-211t48-189t47-248t46-143t45-93t44-130t43-98t42-60t41-73t40-36t39-24t38-40t37-29t36-9t35-10t34-4t33-3t32-6t31-4t30)*y3+(-t176-2t175-t174-4t171-15t170-18t169-7t168-8t167-22t166-44t165-76t164-68t163-50t162-112t161-170t160-179t159-221t158-229t157-294t156-516t155-552t154-459t153-603t152-747t151-920t150-1234t149-1147t148-1048t147-1471t146-1648t145-1682t144-1964t143-1859t142-1900t141-2375t140-2239t139-2042t138-2216t137-2038t136-2048t135-2070t134-1562t133-1323t132-964t131-475t130-456t129+391t128+1258t127+1266t126+2435t125+3114t124+2951t123+5106t122+6138t121+5176t120+7034t119+7971t118+7363t117+10346t116+10869t115+8630t114+11406t113+12631t112+10651t111+13196t110+13346t109+10603t108+13880t107+14625t106+10952t105+12903t104+13448t103+10505t102+12854t101+12885t100+8941t99+10684t98+11409t97+8169t96+9127t95+9157t94+6227t93+7392t92+7740t91+4910t90+5176t89+5463t88+3616t87+3957t86+4021t85+2316t84+2392t83+2648t82+1613t81+1530t80+1563t79+858t78+851t77+950t76+504t75+398t74+439t73+238t72+203t71+219t70+96t69+61t68+82t67+44t66+24t65+24t64+8t63+4t62+8t61+4t60)*x2+(2t176+4t175+2t174+8t171+32t170+40t169+16t168+16t167+44t166+96t165+186t164+180t163+118t162+240t161+384t160+462t159+656t158+670t157+718t156+1288t155+1532t154+1460t153+1992t152+2280t151+2652t150+3964t149+4122t148+3848t147+5344t146+6072t145+6580t144+8744t143+8712t142+8488t141+11586t140+12414t139+12326t138+15212t137+15070t136+15258t135+19966t134+19848t133+18310t132+22004t131+22204t130+22400t129+27018t128+24910t127+22762t126+28176t125+28174t124+26136t123+28748t122+26158t121+25634t120+31660t119+28988t118+24216t117+26638t116+25814t115+25638t114+28598t113+23598t112+20196t111+24538t110+23484t109+20532t108+20652t107+17342t106+17176t105+20752t104+17190t103+13278t102+14304t101+13360t100+13140t99+13782t98+10038t97+8454t96+10636t95+9498t94+7746t93+7442t92+5764t91+5712t90+6968t89+5294t88+3704t87+4056t86+3638t85+3312t84+3470t83+2420t82+1754t81+2266t80+2028t79+1432t78+1360t77+1068t76+828t75+1010t74+828t73+456t72+456t71+442t70+306t69+292t68+228t67+110t66+122t65+130t64+68t63+44t62+40t61+20t60+18t59+18t58+6t57+2t56+4t55+2t54)*xy+(-t176-2t175-t174-4t171-16t170-20t169-8t168-8t167-22t166-48t165-93t164-90t163-59t162-120t161-192t160-231t159-328t158-335t157-359t156-644t155-766t154-730t153-996t152-1140t151-1326t150-1982t149-2061t148-1924t147-2672t146-3036t145-3290t144-4372t143-4356t142-4244t141-5793t140-6207t139-6163t138-7606t137-7535t136-7629t135-9983t134-9924t133-9155t132-11002t131-11102t130-11200t129-13509t128-12455t127-11381t126-14088t125-14087t124-13068t123-14374t122-13079t121-12817t120-15830t119-14494t118-12108t117-13319t116-12907t115-12819t114-14299t113-11799t112-10098t111-12269t110-11742t109-10266t108-10326t107-8671t106-8588t105-10376t104-8595t103-6639t102-7152t101-6680t100-6570t99-6891t98-5019t97-4227t96-5318t95-4749t94-3873t93-3721t92-2882t91-2856t90-3484t89-2647t88-1852t87-2028t86-1819t85-1656t84-1735t83-1210t82-877t81-1133t80-1014t79-716t78-680t77-534t76-414t75-505t74-414t73-228t72-228t71-221t70-153t69-146t68-114t67-55t66-61t65-65t64-34t63-22t62-20t61-10t60-9t59-9t58-3t57-t56-2t55-t54)*y2+(-t224-2t223-t222-5t219-21t218-27t217-11t216-10t215-30t214-80t213-166t212-162t211-101t210-205t209-365t208-524t207-769t206-751t205-816t204-1538t203-1977t202-2140t201-2913t200-3325t199-4197t198-6420t197-6908t196-7169t195-10242t194-12094t193-14090t192-18688t191-19393t190-21462t189-30122t188-33011t187-35132t186-44709t185-47678t184-53356t183-69231t182-70758t181-73563t180-93906t179-99433t178-105554t177-128535t176-129007t175-135315t174-168861t173-170684t172-171897t171-205460t170-206998t169-213086t168-253698t167-246589t166-243500t165-290244t164-287297t163-281931t162-324536t161-312225t160-305313t159-356309t158-340189t157-319957t156-365634t155-350705t154-332221t153-375160t152-349175t151-321135t150-365575t149-343722t148-309991t147-343204t146-317662t145-285606t144-318521t143-290910t142-251461t141-277555t140-255659t139-220145t138-238821t137-214524t136-179990t135-197194t134-178202t133-145046t132-154798t131-138875t130-112368t129-120014t128-105963t127-81988t126-86923t125-77724t124-59326t123-61275t122-53591t121-39727t120-41386t119-36411t118-25810t117-25964t116-22853t115-16048t114-16013t113-13828t112-9109t111-8968t110-7946t109-5121t108-4788t107-4138t106-2549t105-2401t104-2112t103-1204t102-1045t101-938t100-532t99-446t98-388t97-192t96-153t95-147t94-71t93-45t92-42t91-19t90-12t89-12t88-4t87-t86-2t85-t84)*x+(t224+2t223+t222+5t219+21t218+27t217+11t216+10t215+30t214+80t213+166t212+162t211+101t210+205t209+365t208+524t207+769t206+751t205+816t204+1538t203+1977t202+2140t201+2913t200+3325t199+4197t198+6420t197+6908t196+7169t195+10242t194+12094t193+14090t192+18688t191+19393t190+21462t189+30122t188+33011t187+35132t186+44709t185+47678t184+53356t183+69231t182+70758t181+73563t180+93906t179+99433t178+105554t177+128535t176+129007t175+135315t174+168861t173+170684t172+171897t171+205460t170+206998t169+213086t168+253698t167+246589t166+243500t165+290244t164+287297t163+281931t162+324536t161+312225t160+305313t159+356309t158+340189t157+319957t156+365634t155+350705t154+332221t153+375160t152+349175t151+321135t150+365575t149+343722t148+309991t147+343204t146+317662t145+285606t144+318521t143+290910t142+251461t141+277555t140+255659t139+220145t138+238821t137+214524t136+179990t135+197194t134+178202t133+145046t132+154798t131+138875t130+112368t129+120014t128+105963t127+81988t126+86923t125+77724t124+59326t123+61275t122+53591t121+39727t120+41386t119+36411t118+25810t117+25964t116+22853t115+16048t114+16013t113+13828t112+9109t111+8968t110+7946t109+5121t108+4788t107+4138t106+2549t105+2401t104+2112t103+1204t102+1045t101+938t100+532t99+446t98+388t97+192t96+153t95+147t94+71t93+45t92+42t91+19t90+12t89+12t88+4t87+t86+2t85+t84)*y drawTropicalCurve(newfnox,"max");