/////////////////////////////////// // 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) ///////////////////////////// // Case 5: [0, -1, -2, -3] // ///////////////////////////// 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 = (3*t+t^7); poly B34 = (5*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+10t9+181t8+432t7+283t6+24t4+30t3+16t2-1)*x4+(-t28+2t26-t24-10t23-350t22-874t21-222t20+884t19+380t18-2222t17-6789t16-6444t15+2019t14+5552t13-2469t12-15938t11-22989t10-15662t9-4376t8+462t7+126t6-210t5-20t4+30t3+18t2)*x3+(169t36+442t35-49t34-884t33-410t32+2132t31+6737t30+6504t29-3992t28-10856t27-313t26+28018t25+56942t24+38864t23-20456t22-42536t21+5008t20+105904t19+172359t18+100988t17-23095t16-61628t15-4496t14+101306t13+142986t12+64370t11-10747t10-16806t9-5679t8-540t7-387t6-270t5-81t4)*x2+y2+(169t40+442t39+289t38-338t36+806t35+5870t34+9884t33+9071t32+4294t31-1214t30+13724t29+53945t28+86716t27+95727t26+67180t25+29050t24+73230t23+186033t22+301164t21+360657t20+287142t19+170574t18+155880t17+225198t16+354114t15+441936t14+372744t13+244872t12+113832t11+23409t10)*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+10t9+181t8+432t7+283t6+25t4+30t3+18t2)*x4+(-t28+2t26-t24-10t23-350t22-874t21-222t20+884t19+378t18-2222t17-6791t16-6444t15+2021t14+5542t13-2479t12-15958t11-23007t10-15672t9-4376t8+432t7+114t6-270t5-56t4)*x3+(-2t2-2)*x2y+(169t36+442t35-49t34-884t33-409t32+2132t31+6737t30+6504t29-3994t28-10846t27-301t26+28028t25+56949t24+38854t23-20449t22-42456t21+5106t20+106024t19+172444t18+100958t17-22990t16-61418t15-4124t14+101756t13+143307t12+64460t11-10522t10-16626t9-5202t8)*x2+(2t16-2t12+10t11+12t10+10t9+6t8-6t6+30t5+18t4+30t3+18t2)*xy+y2+(169t40+442t39+289t38-338t36+806t35+5870t34+9884t33+9071t32+4294t31-1214t30+13724t29+53945t28+86716t27+95727t26+67180t25+29050t24+73230t23+186033t22+301164t21+360657t20+287142t19+170574t18+155880t17+225198t16+354114t15+441936t14+372744t13+244872t12+113832t11+23409t10)*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; // 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drawTropicalCurve(newfnox,"max");