/////////////////////////////////// // Modifications for Theta Graph // /////////////////////////////////// // The following contains an example for Cell8 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 8: [0, -1, -2, -2] // ///////////////////////////// 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^1+t^5); poly B34 = (5*t^2+t^10); poly B2 = (17*t^2+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+11t12+8t11+2t10+34t8+10t7+16t6+313t4+40t3+30t2-1)*x4+(-t32-t30-34t28-2t27-8t26-2t25-298t24-76t23-26t22-24t21-340t20-858t19-164t18-102t17-2920t16-2682t15-1315t14-232t13-897t12-4232t11-5826t10-460t9-7101t8-11510t7-8972t6-560t5+122t4+40t3+32t2)*x3+(t42+42t38+2t37-t36+584t34+92t33-36t32-2t31+3199t30+1494t29-299t28-82t27+9785t26+10610t25+233t24-984t23+28736t22+37696t21+4544t20-3764t19+53300t18+82894t17+30149t16-7438t15+62257t14+151252t13+62437t12-16770t11+85078t10+160720t9+46951t8-12840t7-10160t6-640t5-256t4)*x2+y2+(t46+2t44+43t42+2t41+84t40+4t39+629t38+94t37+1175t36+184t35+3865t34+1592t33+6603t32+3000t31+13795t30+12288t29+21817t28+21576t27+41887t26+50334t25+68293t24+79092t23+106563t22+128178t21+166178t20+177264t19+205473t18+249112t17+262192t16+320960t15+352376t14+345440t13+387872t12+369920t11+263568t10+184960t9+73984t8)*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+11t12+8t11+2t10+34t8+10t7+16t6+314t4+40t3+32t2)*x4+(-t32-t30-34t28-2t27-8t26-2t25-298t24-76t23-26t22-24t21-340t20-860t19-164t18-106t17-2920t16-2692t15-1317t14-248t13-901t12-4250t11-5844t10-480t9-7133t8-11560t7-9020t6-640t5+58t4)*x3+(-2t2-2)*x2y+(t42+42t38+2t37-t36+585t34+92t33-34t32-2t31+3208t30+1496t29-283t28-78t27+9819t26+10636t25+286t24-936t23+28844t22+37826t21+4721t20-3552t19+53597t18+83248t17+30631t16-6942t15+62834t14+151980t13+63189t12-15810t11+86190t10+161840t9+48263t8-11560t7-9248t6)*x2+(2t17+2t15+8t13+2t12+8t11+2t10+10t9+16t8+10t7+16t6+40t5+32t4+40t3+32t2)*xy+y2+(t46+2t44+43t42+2t41+84t40+4t39+629t38+94t37+1175t36+184t35+3865t34+1592t33+6603t32+3000t31+13795t30+12288t29+21817t28+21576t27+41887t26+50334t25+68293t24+79092t23+106563t22+128178t21+166178t20+177264t19+205473t18+249112t17+262192t16+320960t15+352376t14+345440t13+387872t12+369920t11+263568t10+184960t9+73984t8)*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");