/////////////////////////////////// // Modifications for Theta Graph // /////////////////////////////////// // The following contains an example for Cell7_1 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: // PiecesTypeIICone[1][0].Hrepresentation() // (An inequality (1, -1, 0, 0) x + 0 >= 0, // An inequality (0, 0, -1, 1) x + 0 >= 0, // An inequality (-1, 2, 0, -1) x + 0 >= 0) // We want a cancellation, so we need in(b4^2) = in(b2*b5). /////////////////////////////// // Case 7_1: [0, -1, -4, -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 = (2*t^1+t^5); poly B34 = (t^4+t^10); poly B2 = (4*t^2+t^16); 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+(t32+t20+8t18+2t15+2t14+4t11+2t10+2t9+t8+8t6+4t5+15t4+6t2-1)*x4+(-t52-2t47-2t46-4t43-2t42-2t41-t40-16t38-4t37+t36-6t34-16t33-15t32-t30-32t29-16t28-16t27-12t26-2t25-81t24-32t23+6t22-12t21-56t20-32t19-23t18-20t17-12t16-70t15-32t14-38t13-35t12-16t11-120t10-58t9+t8-8t7-80t6+4t5+16t4+8t2)*x3+(t62+4t58+2t57+t56+2t54+12t53+8t52-t50+20t49+20t48+6t47+6t45+51t44+32t43-6t41+31t40+104t39+48t38-4t37-8t36+160t35+103t34+48t33-2t32+48t31+211t30+158t29-58t28-52t27+140t26+242t25-19t24-58t23-39t22+280t21+101t20+34t19-37t18+32t17+256t16+188t15-184t14-160t13+176t12+88t11-324t10-96t9-48t8-16t7-160t6-16t4)*x2+y2+(t66+2t64+5t62+2t61+10t60+4t59+12t58+14t57+19t56+26t55+23t54+40t53+35t52+62t51+53t50+64t49+88t48+84t47+136t46+96t45+160t44+152t43+220t42+240t41+232t40+336t39+264t38+496t37+344t36+512t35+464t34+576t33+608t32+576t31+704t30+544t29+848t28+672t27+752t26+768t25+688t24+992t23+720t22+1024t21+768t20+1088t19+896t18+1024t17+1024t16+640t15+1088t14+512t13+768t12+256t11+512t10+256t8)*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+(t32+t20+8t18+2t15+2t14+4t11+2t10+2t9+t8+8t6+4t5+16t4+8t2)*x4+(-t52-2t47-2t46-4t43-2t42-2t41-t40-16t38-4t37+t36-6t34-16t33-15t32-t30-32t29-16t28-16t27-12t26-2t25-81t24-32t23+6t22-12t21-56t20-34t19-23t18-24t17-12t16-76t15-34t14-48t13-39t12-24t11-130t10-64t9-15t8-16t7-96t6)*x3+(-2t2-2)*x2y+(t62+4t58+2t57+t56+2t54+12t53+8t52-t50+20t49+20t48+6t47+6t45+51t44+32t43-6t41+31t40+104t39+48t38-4t37-8t36+160t35+104t34+48t33+48t31+216t30+160t29-48t28-48t27+152t26+256t25-32t23-16t22+320t21+128t20+96t19+96t17+304t16+256t15-128t14-96t13+240t12+128t11-256t10-64t9-128t6)*x2+(2t17+2t15+4t13+2t12+6t11+2t10+2t9+8t8+4t7+8t6+4t5+8t4+8t2)*xy+y2+(t66+2t64+5t62+2t61+10t60+4t59+12t58+14t57+19t56+26t55+23t54+40t53+35t52+62t51+53t50+64t49+88t48+84t47+136t46+96t45+160t44+152t43+220t42+240t41+232t40+336t39+264t38+496t37+344t36+512t35+464t34+576t33+608t32+576t31+704t30+544t29+848t28+672t27+752t26+768t25+688t24+992t23+720t22+1024t21+768t20+1088t19+896t18+1024t17+1024t16+640t15+1088t14+512t13+768t12+256t11+512t10+256t8)*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");