//////////////////////////////////////////////////////////////////////// // Refined Modifications for Type IV Graph with Vertical Modification // //////////////////////////////////////////////////////////////////////// // The following contains an example for the refined 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). // We replace our coefficients a by a + t^e for a suitable e. This refinement gives a planar faithful re-embedding of the extended skeleton with the caveat that the branches for a5 and a4 are not separated. 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*(1+t^8)*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+2t35-t34+2t33-8t32-2t31-22t30-7t28-8t27-10t26-2t25+2t24-6t23-28t22+t20-2t19-24t18-8t17+t16-4t15-28t14+4t13-11t10-6t9+6t8+12t5+9)*x3+(-2t5-2)*x2y+(t66+2t64+3t62+4t60+6t58+11t56+14t54-6t53+30t52-6t51+19t50-10t49+30t48+12t47+24t46-16t45+60t44+6t43+24t42-12t41+34t40+58t39+3t38-18t37+33t36+48t35+29t34-18t33+27t32+78t31+24t30+9t28+72t27+39t26+36t23+36t22+18t18)*x2+(2t30+2t26+2t25+8t22+2t21+8t18+8t17+6t14+8t13+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*(1+t^8)*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+2t35+3t34+2t33+22t32-2t31+58t30+29t28-8t27+18t26-2t25-70t24-6t23+20t22+t20-2t19+36t18-8t17-323t16-4t15+44t14+4t13+25t10-6t9-642t8+12t5-477)*x3+(-2t5-2)*x2y+(-2t66-t64-5t60-18t58+11t56+32t54-6t52-41t50-4t49+137t48+6t47+267t46+20t45+120t44+18t43+18t41+310t40+4t39+639t38+36t37+396t36-48t35+146t34-24t33-204t32-96t31+462t30+12t29+324t28-180t27+276t26-216t25-2070t24-198t23+72t22+108t21+27t20-54t19+306t18-324t17-4698t16-108t15+216t14+324t13+189t10-162t9-5508t8+324t5-2673)*x2+(2t30+2t26+2t25+8t22-2t21+8t18+8t17-4t16+6t14-16t13+6t10+6t9-24t8-30t5-36)*xy+y2+(-t82-t80-t78-2t76-16t74-5t72+2t71-4t70-2t69-9t68-90t66-6t65+62t64+36t63+77t62-4t61+76t60+26t59-215t58-30t57+583t56+224t55+625t54+48t53+696t52+156t51-140t50-60t49+1755t48+570t47+1590t46+192t45+2133t44+126t43+258t42-414t41+1602t40+288t39+819t38+342t37+2970t36-702t35+279t34-1998t33-3159t32-1242t31-2538t30+1080t29+1701t28-1296t27-378t26-4050t25-11502t24-2106t23-3564t22+3240t21+243t20-486t19-324t18-3888t17-16767t16-972t15-972t14+4860t13+243t10-1458t9-13122t8+2916t5-4374)*x+(2t46+2t42+2t41+20t38+20t34+20t33-2t32+72t30-4t29+72t26+72t25-24t24+108t22-36t21+108t18+108t17-108t16+54t14-108t13+54t10+54t9-216t8-108t5-162)*y+(t92+2t88+2t87+21t84+2t83+t82+40t80+40t79-2t78+192t76+36t75+18t74-2t73+344t72+340t71-48t70+1000t68+268t67+128t66-44t65+1657t64+1580t63-496t62+4t61+3258t60+1044t59+412t58-420t57+4884t56+4248t55-2880t54+84t53+6858t52+2160t51+234t50-2268t49+9108t48+6156t47-10260t46+756t45+9288t44+1836t43-2592t42-7560t41+11232t40+2700t39-22896t38+3780t37+7776t36-972t35-8856t34-15876t33+11502t32-4860t31-31104t30+11340t29+3645t28-2916t27-12636t26-20412t25+15066t24-7290t23-23328t22+20412t21+729t20-1458t19-8019t18-14580t17+20412t16-2916t15-7290t14+20412t13-1458t10-4374t9+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*(1+t^8)*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; // 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drawTropicalCurve(newfnox,"max");