/////////////////////////////////// // 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) ///////////////////////////// // Case 6: [0, -2, -3, -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 = (4*t^2+t^5); poly B34 = (5*t^3+t^10); poly B2 = (17*t^3+t^6); 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+(t20+2t15+10t13+9t12+2t10+34t9+10t8+16t7+314t6+40t5+31t4-2t2-1)*x4+(-t32-t30-34t29-10t27-289t26-12t25-91t24-10t23-364t22-850t21-90t20-3068t19-2347t18-542t17-1298t16-948t15-4326t14-5868t13-7682t12-11732t11-9312t10-534t9+414t8+96t7+378t6+40t5+32t4)*x3+(t42+42t39+2t37+576t36+10t35+89t34+2822t33+418t32+1412t31+4344t30+5754t29+9683t28+416t27+28650t26+32264t25+6535t24+49604t23+48522t22+44094t21+30612t20+66248t19+151173t18+74280t17+97896t16+160350t15+44580t14-30184t13-27257t12-13096t11-10160t10-640t9-256t8)*x2+y2+(t46+2t44+42t43+t42+86t41+577t40+56t39+1246t38+2878t37+1182t36+7212t35+5568t34+11656t33+20042t32+13590t31+54829t30+39886t29+76781t28+121136t27+95806t26+193554t25+146988t24+229064t23+285305t22+277128t21+473264t20+421736t19+465664t18+452608t17+263568t16+184960t15+73984t14)*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+(t20+2t15+10t13+9t12+2t10+34t9+10t8+16t7+314t6+40t5+32t4)*x4+(-t32-t30-34t29-10t27-289t26-12t25-91t24-10t23-364t22-850t21-90t20-3070t19-2347t18-546t17-1306t16-950t15-4344t14-5868t13-7704t12-11748t11-9334t10-606t9+372t8+314t6)*x3+(-2t2-2)*x2y+(t42+42t39+2t37+576t36+10t35+90t34+2822t33+420t32+1420t31+4345t30+5772t29+9699t28+438t27+28706t26+32286t25+6680t24+49710t23+48718t22+44462t21+30866t20+66816t19+151585t18+74896t17+98721t16+161126t15+46036t14-28832t13-25721t12-11560t11-9248t10)*x2+(2t17+2t15+8t14+10t12+12t10+16t9+10t8+56t7+32t6+40t5+32t4)*xy+y2+(t46+2t44+42t43+t42+86t41+577t40+56t39+1246t38+2878t37+1182t36+7212t35+5568t34+11656t33+20042t32+13590t31+54829t30+39886t29+76781t28+121136t27+95806t26+193554t25+146988t24+229064t23+285305t22+277128t21+473264t20+421736t19+465664t18+452608t17+263568t16+184960t15+73984t14)*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");