////////////////////////////////////////////// // Refined Modifications for Dumbbell Graph // ////////////////////////////////////////////// // The following contains an example for the modified tropical dumbbell graph in R^3. 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. // We replace our coefficients a and b by a + t^e and b+t^e', respectively. This refinement gives a planar faithful re-embedding of the extended skeleton. 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^10+t^12); poly b3 = (t^5+t^9); poly b4 = (3*t^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; // A5*A4*A3; // (-t^44-2*t^40-8*t^39-t^36-16*t^35-22*t^34-8*t^31-44*t^30-24*t^29-22*t^26-48*t^25-9*t^24-24*t^21-18*t^20-9*t^16) // A5*A4*A2; // (-t^50-2*t^48-t^46-8*t^45-16*t^43-8*t^41-22*t^40-44*t^38-22*t^36-24*t^35-48*t^33-24*t^31-9*t^30-18*t^28-9*t^26) map P2 = rr, A2, A3, A4, A5, x, y, z; poly f27 = P2(f); ring r = (0,t),(x,y),dp; map newP2 = rr, 0,0,0,0,x,y,0; poly f2 = newP2(f27); // f2; // -x5+(t24+2t22+t20+t18+t16+2t14+6t11+9t6-2t5-1)*x4+(-t42-3t40-5t38-5t36-6t35-3t34-12t33-6t31-11t30-4t29-17t28+4t27-9t26-10t25-6t24+2t23+2t22+2t21-16t20+4t19+t18+13t16+2t15+2t14+24t11+t10+9t6)*x3+(t58+2t56+3t54+6t53+3t52+12t51+18t49+6t48+22t47+14t46+6t45+23t44-10t43+33t42-10t41-14t39-29t38-4t37-18t36-42t35-25t34-48t33-2t32-32t31-54t30-24t29-18t28-31t26-48t25-9t24-24t21-18t20-9t16)*x2+y2+(t68+2t66+3t64+8t63+4t62+16t61+3t60+24t59+24t58+32t57+45t56+24t55+66t54+40t53+88t52+56t51+66t50+72t49+53t48+96t47+40t46+72t45+27t44+48t43+36t42+24t41+27t40+18t38+9t36)*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^4)*x-b5*(1+t^2)*x^2); // g2; // -x5+(t24+2t22+t20+t18+t16+3t14+2t12+6t11+t10+2t9+4t7+9t6+t4+2t2)*x4+(-t42-3t40-5t38-5t36-6t35-3t34-14t33-8t31-11t30-8t29-27t28-19t26-12t25-26t24-14t23-18t22-12t21-26t20-24t19-15t18-28t17+7t16-12t15-10t14-14t13-12t12+24t11-5t10-6t8+9t6)*x3+(-2t7-2t5-2t2-2)*x2y+(t58+2t56+3t54+6t53+4t52+12t51+18t49+10t48+30t47+14t46+6t45+29t44+22t43+55t42-10t41+4t40+34t39+59t38+20t37-17t36-10t35+107t34+48t33+7t32-24t31+34t30+120t29+18t28-9t26+48t25+45t24+18t20)*x2+(2t26+4t22+8t21+2t18+16t17+6t16+8t13+12t12+6t8)*xy+y2+(t68+2t66+3t64+8t63+4t62+16t61+3t60+24t59+24t58+32t57+45t56+24t55+66t54+40t53+88t52+56t51+66t50+72t49+53t48+96t47+40t46+72t45+27t44+48t43+36t42+24t41+27t40+18t38+9t36)*x drawTropicalCurve(g2,"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^4)*x-b5*(1+t^2)*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");