//////////////////////////////////////////////////////////////// // Modifications for Type IV Graph with Vertical modification // //////////////////////////////////////////////////////////////// // The following contains an example for the modified Type IV graph in R^4 (with one smooth genus 1 component and one rational component with a node). The curve can be visualized by means of projections. // Since our modifications involve square roots, we take our branch points to be squares, resp. negative of squares, for computational convenience. // The parameters w2,..., w5 satisfy w2 < w3 < w4 = w5 with val(a5-a4)=val(a5) = val(a4). LIB "all.lib"; LIB "poly.lib"; LIB "tropical.lib"; LIB "elim.lib"; ring rr = (0,t), (a2,a3,a4,a5, x,y,z),dp; poly f=y^2-x*(x-a2)*(x-a3)*(x-a4)*(x-a5); // f; // -a2*a3*a4*a5*x+a2*a3*a4*x^2+a2*a3*a5*x^2+a2*a4*a5*x^2+a3*a4*a5*x^2-a2*a3*x^3-a2*a4*x^3-a3*a4*x^3-a2*a5*x^3-a3*a5*x^3-a4*a5*x^3+a2*x^4+a3*x^4+a4*x^4+a5*x^4-x^5+y^2 setring(rr); poly b2 = (t^14+t^16); poly b3 = (t^5+t^9); poly b4 = (3+t^8); poly b5 = (1+t^5); poly A2 = b2^2; poly A3 = b3^2; poly A4 = b4^2; poly A5 = -1*b5^2; map P2 = rr, A2, A3, A4, A5, 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+(t32+2t30+t28+t18+t16+2t14+6t8-2t5+8)*x4+(-t50-3t48-5t46-5t44-2t42-6t40-12t38+2t37-6t36+4t35-t34+2t33-8t32-18t30-7t28-6t26+2t24+2t23-12t22+2t21+t20+4t19-8t18+t16+2t15-16t14+12t13+t10+6t8+18t5+9)*x3+(t66+2t64+3t62+3t60+6t58+9t56-2t55+14t54-6t53+21t52-10t51+18t50-10t49+14t48-6t47+22t46-16t45+30t44-26t43+15t42-12t41-10t40-2t39-13t38-18t37-13t36-40t35-t34-18t33-21t32-14t31-20t30-24t28-24t27-7t26-18t24-30t23-12t22-9t20-36t19-15t18-18t15-18t14-9t10)*x2+y2+(t76+2t74+3t72+2t71+4t70+4t69+9t68+6t67+15t66+8t65+21t64+18t63+27t62+28t61+31t60+38t59+39t58+48t57+47t56+54t55+55t54+60t53+51t52+66t51+45t50+72t49+39t48+54t47+33t46+36t45+36t44+18t43+27t42+18t40+9t38)*x drawTropicalCurve(f2,"max"); /////////////////// // XZ-projection // /////////////////// poly b2 = newP2(b2); poly b3 = newP2(b3); poly b4 = newP2(b4); poly b5 = newP2(b5); poly g2 = substitute(f2, y, y+b3*b4*b5*x-b5*x^2); // g2; // -x5+(t32+2t30+t28+t18+t16+2t14+t10+6t8+9)*x4+(-t50-3t48-5t46-5t44-2t42-6t40-12t38+2t37-6t36+4t35-t34+2t33-8t32-18t30-7t28-2t27-6t26+2t24-16t22+2t21+t20-2t19-12t18-2t17+t16-4t15-28t14+10t13-11t10-6t9+6t8+12t5+9)*x3+(-2t5-2)*x2y+(t66+2t64+3t62+3t60+6t58+9t56-2t55+14t54-6t53+21t52-10t51+18t50-10t49+14t48-6t47+22t46-16t45+31t44-26t43+15t42-12t41-8t40-13t38-18t37-6t36-36t35-18t33-9t32-18t30-9t28)*x2+(2t22+2t18+2t17+6t14+2t13+6t10+6t9+6t5)*xy+y2+(t76+2t74+3t72+2t71+4t70+4t69+9t68+6t67+15t66+8t65+21t64+18t63+27t62+28t61+31t60+38t59+39t58+48t57+47t56+54t55+55t54+60t53+51t52+66t51+45t50+72t49+39t48+54t47+33t46+36t45+36t44+18t43+27t42+18t40+9t38)*x drawTropicalCurve(g2,"max"); /////////////////// // UY-projection // /////////////////// poly guy = substitute(f2, x, x + b4^2); // guy; // -x5+(t32+2t30+t28+t18-4t16+2t14-24t8-2t5-37)*x4+(-t50+t48+3t46-t44-2t42+18t40+36t38+2t37+18t36+4t35+3t34+2t33+22t32+62t30+29t28+18t26-70t24+2t23+36t22-6t21+t20+4t19+28t18-327t16+2t15+56t14-36t13+t10-666t8-54t5-513)*x3+(-2t66-t64-6t60-18t58+9t56-2t55+32t54-15t52+2t51-42t50-4t49+121t48-6t47+277t46+20t45+90t44+46t43-3t42+24t41+266t40+4t39+743t38+30t37+350t36+80t35+164t34+36t33-258t32+28t31+850t30-72t29+291t28+48t27+338t26-2160t24+60t23+672t22-324t21+18t20+72t19+273t18-5022t16+36t15+522t14-648t13+18t10-6156t8-486t5-3159)*x2+y2+(-t82-t80-t78-4t76-16t74-9t72-2t71-4t70-2t69-39t68-2t67-92t66-6t65+6t64-18t63+85t62-4t61-110t60+10t59-237t58-24t57+267t56-52t55+761t54+46t53+82t52+154t51-206t50+36t49+839t48-28t47+2534t46+144t45+975t44+462t43+364t42+216t41+126t40+132t39+4227t38-108t37+1728t36+540t35+1281t34+162t33-4779t32+324t31+4158t30-1080t29+999t28+432t27+1620t26-13986t24+378t23+3132t22-2430t21+81t20+324t19+918t18-21627t16+162t15+1620t14-2916t13+81t10-18954t8-1458t5-7290)*x drawTropicalCurve(guy,"max"); /////////////////// // UZ-projection // /////////////////// poly guz = substitute(substitute(f2, y, y+b3*b4*b5*x-b5*x^2),x, x + b4^2); // guz; // -x5+(t32+2t30+t28+t18-4t16+2t14+t10-24t8-36)*x4+(-t50+t48+3t46-t44-2t42+18t40+36t38+2t37+18t36+4t35+3t34+2t33+22t32+62t30+29t28-2t27+22t26-70t24+32t22+2t21+t20-2t19+48t18-2t17-323t16-4t15+44t14+10t13+25t10-6t9-642t8+12t5-477)*x3+(-2t5-2)*x2y+(-2t66-t64-6t60-18t58+9t56-2t55+32t54-15t52+2t51-42t50-4t49+121t48-6t47+277t46+20t45+91t44+40t43+3t42+24t41+268t40+731t38+42t37+357t36+30t35+225t34+30t33-240t32-12t31+744t30+66t29+306t28-90t27+561t26-54t25-2070t24-72t23+360t22+270t21+27t20-54t19+612t18-162t17-4698t16-108t15+216t14+486t13+189t10-162t9-5508t8+324t5-2673)*x2+(2t22-4t21+2t18+2t17-4t16+6t14-22t13+6t10+6t9-24t8-30t5-36)*xy+y2+(-t82-t80-t78-4t76-16t74-9t72-2t71-4t70-2t69-39t68-2t67-92t66-6t65+6t64-18t63+85t62-4t61-108t60+4t59-233t58-24t57+271t56-54t55+749t54+54t53+108t52+72t51-144t50+30t49+891t48-66t47+2358t46+282t45+1107t44+18t43+750t42+126t41+414t40-144t39+3195t38+882t37+2052t36-648t35+2493t34-378t33-3807t32-648t31+1134t30+2700t29+1377t28-1134t27+3564t26-1620t25-11502t24-1296t23-1296t22+5670t21+243t20-486t19+2268t18-2430t17-16767t16-972t15-972t14+6318t13+243t10-1458t9-13122t8+2916t5-4374)*x+(2t38-2t37+2t34+2t33-2t32+18t30-22t29+18t26+18t25-24t24+54t22-90t21+54t18+54t17-108t16+54t14-162t13+54t10+54t9-216t8-108t5-162)*y+(t76-2t75+t74+2t72-4t70+2t69+19t68-38t67+21t66-2t65+37t64-4t63-82t62+46t61+153t60-306t59+187t58-42t57+294t56-72t55-720t54+462t53+675t52-1350t51+909t50-378t49+1332t48-540t47-3510t46+2646t45+1755t44-3510t43+2565t42-1890t41+3942t40-2160t39-10260t38+9450t37+2673t36-5346t35+4023t34-5670t33+8586t32-4860t31-17982t30+21546t29+2187t28-4374t27+2673t26-10206t25+15066t24-5832t23-17496t22+30618t21+729t20-1458t19-729t18-10206t17+20412t16-2916t15-7290t14+24786t13-1458t10-4374t9+17496t8+8748t5+6561) drawTropicalCurve(guz,"max"); /////////////////// // ZY-projection // /////////////////// ring s = (0,t),(x,y,z),dp; map P = r, x,y; poly ff= P(f2); poly b3 = P(b3); poly b4 = P(b4); poly b5 = P(b5); 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+(t69+4t67+6t65+t64+4t63+4t62+t61+6t60+4t58+t56-t54-2t52-2t50-t49-2t48-4t46+2t45-6t44-3t41+5t40-6t39+2t38+5t37-2t36+5t35-t34+10t33-3t32-t31-5t30+19t29-18t27-3t26-6t25+7t24-4t23-39t22+51t21-8t20-33t19-t18-31t17+56t16-15t15-59t14+113t13-7t10-39t9+120t8+96t5+99)*x4+(-4t69-16t67-24t65-4t64-16t63-16t62-4t61-24t60-16t58-4t56+4t54+8t52+8t50+4t49+8t48+4t47+16t46+24t44+4t43+12t42+12t41+4t40+24t39+4t38-8t37+8t36+4t35+4t34-24t33+6t32+8t31+28t30-68t29+6t28+42t27+24t26+28t25-34t24+22t23+126t22-192t21+14t20+66t19+50t18+114t17-220t16+18t15+154t14-390t13+50t10+126t9-456t8-306t5-360)*x3y+(6t69+24t67+36t65+6t64+24t63+24t62+6t61+36t60+24t58+6t56-6t54-12t52-12t50-6t49-12t48-10t47-24t46-8t45-36t44-10t43-30t42-18t41-30t40-36t39-18t38-12t36-30t35-6t34+20t33-3t32-16t31-50t30+94t29-15t28-33t27-48t26-42t25+57t24-39t23-159t22+276t21+17t20-33t19-121t18-161t17+326t16+55t15-149t14+523t13-57t10-159t9+660t8+401t5+508)*x2y2+(-4t69-16t67-24t65-4t64-16t63-16t62-4t61-24t60-16t58-4t56+4t54+8t52+8t50+4t49+8t48+8t47+16t46+8t45+24t44+8t43+24t42+12t41+28t40+24t39+16t38+4t37+8t36+28t35+4t34-8t33+12t31+36t30-60t29+12t28+12t27+36t26+28t25-40t24+28t23+96t22-180t21-24t20+96t18+104t17-216t16-64t15+72t14-328t13+32t10+96t9-432t8-248t5-328)*xy3+(t69+4t67+6t65+t64+4t63+4t62+t61+6t60+4t58+t56-t54-2t52-2t50-t49-2t48-2t47-4t46-2t45-6t44-2t43-6t42-3t41-7t40-6t39-4t38-t37-2t36-7t35-t34+2t33-3t31-9t30+15t29-3t28-3t27-9t26-7t25+10t24-7t23-24t22+45t21+6t20-24t18-26t17+54t16+16t15-18t14+82t13-8t10-24t9+108t8+62t5+82)*y4+(t110-t109+4t108-5t107+11t106-11t105+22t104-15t103+27t102-10t101+19t100-t99+15t98-13t97+37t96-46t95+41t94-48t93-38t92-8t91-93t90-15t89+34t88-120t87+149t86-120t85+18t84+145t83-58t82+320t81+282t80+109t79+602t78+96t77+418t76+716t75+80t74+1070t73+303t72+598t71+907t70+117t69+880t68+271t67+505t66+731t65+42t64+389t63+135t62-118t61-181t60-770t59+453t58-410t57-888t56-834t55-531t54+382t53-1642t52-752t51+538t50-26t49-1218t48-1266t47+1387t46+952t45-2060t44-1097t43+1302t42+1723t41-2182t40-1625t39+1825t38+530t37-1873t36-1844t35+1644t34+1869t33-4309t32-1605t31+699t30+800t29-1380t28-3696t27+906t26-198t25-3026t24-580t23-36t22+558t21-1169t20-2871t19+1737t18+297t17+54t16+123t15-378t14+945t13+1188t10+135t9+108t8+621t5+81)*x3+(-3t110+3t109-12t108+15t107-33t106+33t105-66t104+45t103-81t102+30t101-57t100+3t99-45t98+39t97-111t96+138t95-123t94+144t93+114t92+24t91+279t90+45t89-102t88+366t87-459t86+378t85-78t84-399t83+162t82-960t81-818t80-363t79-1758t78-276t77-1282t76-2016t75-364t74-3138t73-929t72-1892t71-2433t70-645t69-2464t68-1043t67-1623t66-2061t65-510t64-881t63-581t62+463t60+1904t59-459t58+1154t57+2060t56+3342t55+1741t54+408t53+4510t52+3178t51+1430t50+480t49+4366t48+4684t47-209t46+58t45+5040t44+4083t43+70t42-2141t41+5742t40+3949t39-1631t38+360t37+3331t36+3348t35-2136t34-2583t33+7731t32+747t31-2253t30+114t29+636t28+4302t27-4254t26-288t25+5874t24-3990t23-2448t22-1728t21-453t20+3537t19-7515t18-1053t17-162t16-4329t15-918t14-2997t13-5076t10-567t9-324t8-2025t5-243)*x2y+(3t110-3t109+12t108-15t107+33t106-33t105+66t104-45t103+81t102-30t101+57t100-3t99+45t98-39t97+111t96-138t95+123t94-144t93-114t92-24t91-279t90-45t89+102t88-369t87+465t86-387t85+90t84+381t83-156t82+960t81+804t80+381t79+1734t78+270t77+1296t76+1950t75+426t74+3102t73+939t72+1941t71+2289t70+792t69+2376t68+1158t67+1677t66+1995t65+702t64+738t63+669t62+177t61-423t60-1701t59+9t58-1116t57-1758t56-3762t55-1815t54-1185t53-4302t52-3639t51-2952t50-681t49-4722t48-5127t47-1767t46-1515t45-4470t44-4479t43-2058t42+627t41-5340t40-3486t39-291t38-1335t37-2187t36-2256t35+738t34+1071t33-5133t32+1287t31+2331t30-1371t29+1116t28-909t27+5022t26+729t25-4272t24+6855t23+3726t22+1755t21+2433t20-999t19+8667t18+1134t17+162t16+6309t15+1944t14+3078t13+5832t10+648t9+324t8+2106t5+243)*xy2+(-t110+t109-4t108+5t107-11t106+11t105-22t104+15t103-27t102+10t101-19t100+t99-15t98+13t97-37t96+46t95-41t94+48t93+38t92+8t91+93t90+15t89-34t88+123t87-155t86+129t85-30t84-127t83+52t82-320t81-268t80-127t79-578t78-90t77-432t76-650t75-142t74-1034t73-313t72-647t71-763t70-264t69-792t68-386t67-559t66-665t65-234t64-246t63-223t62-59t61+141t60+567t59-3t58+372t57+586t56+1254t55+605t54+395t53+1434t52+1213t51+984t50+227t49+1574t48+1709t47+589t46+505t45+1490t44+1493t43+686t42-209t41+1780t40+1162t39+97t38+445t37+729t36+752t35-246t34-357t33+1711t32-429t31-777t30+457t29-372t28+303t27-1674t26-243t25+1424t24-2285t23-1242t22-585t21-811t20+333t19-2889t18-378t17-54t16-2103t15-648t14-1026t13-1944t10-216t9-108t8-702t5-81)*y3+(-t148+2t147-5t146+8t145-15t144+17t143-26t142+24t141-38t140+36t139-64t138+74t137-130t136+147t135-215t134+236t133-283t132+309t131-323t130+383t129-349t128+488t127-368t126+674t125-415t124+862t123-383t122+920t121-5t120+513t119+606t118+17t117+782t116-83t115+281t114+94t113-193t112-1432t111+303t110-3855t109+1022t108-5575t107-1147t106-4443t105-5802t104-2495t103-8443t102-3349t101-5189t100-8451t99-368t98-9360t97-622t96+1030t95-5964t94+12088t93-3014t92+11637t91+15961t90+4480t89+32641t88+11701t87+30779t86+31823t85+21865t84+44815t83+28824t82+36696t81+406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drawTropicalCurve(newfnox,"max");