方程组求解（或无解证明）

mathe

我本来想将特殊值Q_Y=1代入的。 不过现在最好能够将 rootOf((3Q_Y^2+1-3Q_Y)_Z^2+(-1+2Q_Y)_Z+1-Q_Y+Q_Y^2) 用符号t替换显示比较好，现在表达式太大了。此外，结果用pdf文件显示比较好看，但是我几乎无法处理它。 另外是否有某个变量的解直接是rootOf((3Q_Y^2+1-3Q_Y)_Z^2+(-1+2Q_Y)_Z+1-Q_Y+Q_Y^2),如果有那么也可以很容易证明这个解必然不是实数解

mathe

另外如下图我们可以有一个17棵树（4棵在无穷远点，就是那些平行线的交点）15行的整数解（或有理解）。如果证明了上面方程没有有理数解，我们就证明了对于17棵树，最优复数解是17行，实数解是16行，而整数解只有15行。

数学星空

对于第三个方程,经过代换后有: (1) RootOf(1-_Z+_Z^2) = A B_Y = -A, D_Y = 1-A, E_X = 1, E_Y = 0, G_X = 1, H_X = 1, H_Y = 0, I_X = 1-A, I_Y = 0, K_Y = A, L_X = 1, L_Y = 0, M_X = A, M_Y = 1, O_X = 1, O_Y = 1, P_X = 1, P_Y = 1-A, Q_X = 1, Q_Y = 0 (2)RootOf(1-_Z+_Z^2) = B B_Y = -B, D_Y = -1, E_X = -1+B, E_Y = 1+B, G_X = -B, H_X = -1+B, H_Y = 2-B, I_X = -1, I_Y = 1+B, K_Y = B, L_X = -B, L_Y = 1+B, M_X = -1, M_Y = 1, O_X = -1+B, O_Y = 1, P_X = -B, P_Y = B, Q_X = -1/2, Q_Y = (1/2)*B+1/2 (3)RootOf((3*Q_Y^2+1-3*Q_Y)*_Z^2+(-1+2*Q_Y)*_Z+1-Q_Y+Q_Y^2) = k D_Y = -(1+1404*Q_Y^5-k+53*Q_Y^2+5130*Q_Y^9+972*Q_Y^11-12*Q_Y-162*Q_Y^12-2808*Q_Y^10-3570*Q_Y^6+5760*Q_Y^7-229*Q_Y^4-76*Q_Y^3-6462*Q_Y^8+673*Q_Y^3*k-150*k*Q_Y^2-891*k*Q_Y^10+19*k*Q_Y+5994*k*Q_Y^7-5670*Q_Y^6*k+3924*Q_Y^5*k-1953*Q_Y^4*k-4617*k*Q_Y^8+2511*k*Q_Y^9+162*Q_Y^11*k)/(162*Q_Y^12*k+54*Q_Y^11-972*Q_Y^11*k-297*Q_Y^10+2916*k*Q_Y^10+873*Q_Y^9-5670*k*Q_Y^9-1701*Q_Y^8+7857*k*Q_Y^8+2382*Q_Y^7-8100*k*Q_Y^7-2478*Q_Y^6+6300*Q_Y^6*k-3672*Q_Y^5*k+1924*Q_Y^5+1557*Q_Y^4*k-1099*Q_Y^4+443*Q_Y^3-450*Q_Y^3*k+78*k*Q_Y^2-116*Q_Y^2+17*Q_Y-6*k*Q_Y-1), E_X = (-27*Q_Y^10+27*k*Q_Y^10+162*Q_Y^9-135*k*Q_Y^9+324*k*Q_Y^8-459*Q_Y^8+816*Q_Y^7-468*k*Q_Y^7-996*Q_Y^6+420*Q_Y^6*k-210*Q_Y^5*k+858*Q_Y^5+14*Q_Y^4*k-518*Q_Y^4+208*Q_Y^3+56*Q_Y^3*k-37*k*Q_Y^2-49*Q_Y^2+5*Q_Y+10*k*Q_Y-k)/((-18*Q_Y^8+72*Q_Y^7-147*Q_Y^6+12*Q_Y^5*k+189*Q_Y^5-160*Q_Y^4-30*Q_Y^4*k+36*Q_Y^3*k+89*Q_Y^3-29*Q_Y^2-24*k*Q_Y^2+8*k*Q_Y+4*Q_Y-k)*(3*Q_Y^2+1-3*Q_Y)), E_Y = -(27*k*Q_Y^7+9*Q_Y^7-108*Q_Y^6*k-9*Q_Y^6-21*Q_Y^5+189*Q_Y^5*k+60*Q_Y^4-195*Q_Y^4*k-70*Q_Y^3+120*Q_Y^3*k-39*k*Q_Y^2+41*Q_Y^2+5*k*Q_Y-11*Q_Y+1)*Q_Y/(-18*Q_Y^8+72*Q_Y^7-147*Q_Y^6+12*Q_Y^5*k+189*Q_Y^5-160*Q_Y^4-30*Q_Y^4*k+36*Q_Y^3*k+89*Q_Y^3-29*Q_Y^2-24*k*Q_Y^2+8*k*Q_Y+4*Q_Y-k), G_X = (48*Q_Y^4*k-44*Q_Y^3*k-30*Q_Y^5*k+9*Q_Y^6*k+25*k*Q_Y^2-8*k*Q_Y-1+5*Q_Y+16*Q_Y^3-15*Q_Y^4-11*Q_Y^2-3*Q_Y^6+9*Q_Y^5+k)/((3*Q_Y^2+1-3*Q_Y)*(6*Q_Y^4*k-12*Q_Y^3*k+12*k*Q_Y^2-6*k*Q_Y-1+3*Q_Y+k-3*Q_Y^2+2*Q_Y^3)), H_X = (-27*Q_Y^10+27*k*Q_Y^10+162*Q_Y^9-135*k*Q_Y^9+324*k*Q_Y^8-459*Q_Y^8+816*Q_Y^7-468*k*Q_Y^7-996*Q_Y^6+420*Q_Y^6*k-210*Q_Y^5*k+858*Q_Y^5+14*Q_Y^4*k-518*Q_Y^4+208*Q_Y^3+56*Q_Y^3*k-37*k*Q_Y^2-49*Q_Y^2+5*Q_Y+10*k*Q_Y-k)/((-18*Q_Y^8+72*Q_Y^7-147*Q_Y^6+12*Q_Y^5*k+189*Q_Y^5-160*Q_Y^4-30*Q_Y^4*k+36*Q_Y^3*k+89*Q_Y^3-29*Q_Y^2-24*k*Q_Y^2+8*k*Q_Y+4*Q_Y-k)*(3*Q_Y^2+1-3*Q_Y)), H_Y = -(81*Q_Y^11+81*Q_Y^11*k-378*Q_Y^10-567*k*Q_Y^10+810*Q_Y^9+1782*k*Q_Y^9-981*Q_Y^8-3483*k*Q_Y^8+558*Q_Y^7+4698*k*Q_Y^7+210*Q_Y^6-4536*Q_Y^6*k+3150*Q_Y^5*k-714*Q_Y^5-1530*Q_Y^4*k+680*Q_Y^4-360*Q_Y^3+486*Q_Y^3*k-89*k*Q_Y^2+109*Q_Y^2-17*Q_Y+7*k*Q_Y+1)*Q_Y/(162*Q_Y^12*k+54*Q_Y^11-972*Q_Y^11*k-297*Q_Y^10+2916*k*Q_Y^10+873*Q_Y^9-5670*k*Q_Y^9-1701*Q_Y^8+7857*k*Q_Y^8+2382*Q_Y^7-8100*k*Q_Y^7-2478*Q_Y^6+6300*Q_Y^6*k-3672*Q_Y^5*k+1924*Q_Y^5+1557*Q_Y^4*k-1099*Q_Y^4+443*Q_Y^3-450*Q_Y^3*k+78*k*Q_Y^2-116*Q_Y^2+17*Q_Y-6*k*Q_Y-1), I_X = (54*k*Q_Y^10+27*Q_Y^9-297*k*Q_Y^9-135*Q_Y^8+783*k*Q_Y^8-1296*k*Q_Y^7+336*Q_Y^7+1470*Q_Y^6*k-534*Q_Y^6-1176*Q_Y^5*k+582*Q_Y^5-444*Q_Y^4+658*Q_Y^4*k-244*Q_Y^3*k+232*Q_Y^3+53*k*Q_Y^2-77*Q_Y^2-5*k*Q_Y+14*Q_Y-1)/((-18*Q_Y^8+72*Q_Y^7-147*Q_Y^6+12*Q_Y^5*k+189*Q_Y^5-160*Q_Y^4-30*Q_Y^4*k+36*Q_Y^3*k+89*Q_Y^3-29*Q_Y^2-24*k*Q_Y^2+8*k*Q_Y+4*Q_Y-k)*(3*Q_Y^2+1-3*Q_Y)), I_Y = -(27*k*Q_Y^7+9*Q_Y^7-108*Q_Y^6*k-9*Q_Y^6-21*Q_Y^5+189*Q_Y^5*k+60*Q_Y^4-195*Q_Y^4*k-70*Q_Y^3+120*Q_Y^3*k-39*k*Q_Y^2+41*Q_Y^2+5*k*Q_Y-11*Q_Y+1)*Q_Y/(-18*Q_Y^8+72*Q_Y^7-147*Q_Y^6+12*Q_Y^5*k+189*Q_Y^5-160*Q_Y^4-30*Q_Y^4*k+36*Q_Y^3*k+89*Q_Y^3-29*Q_Y^2-24*k*Q_Y^2+8*k*Q_Y+4*Q_Y-k), K_Y = (9*Q_Y^4*k+9*k*Q_Y^2-15*Q_Y^3*k+12*Q_Y^3+6*Q_Y-13*Q_Y^2-2*k*Q_Y-3*Q_Y^4-1)/(6*Q_Y^4*k-12*Q_Y^3*k+12*k*Q_Y^2-6*k*Q_Y-1+3*Q_Y+k-3*Q_Y^2+2*Q_Y^3), L_X = (48*Q_Y^4*k-44*Q_Y^3*k-30*Q_Y^5*k+9*Q_Y^6*k+25*k*Q_Y^2-8*k*Q_Y-1+5*Q_Y+16*Q_Y^3-15*Q_Y^4-11*Q_Y^2-3*Q_Y^6+9*Q_Y^5+k)/((3*Q_Y^2+1-3*Q_Y)*(6*Q_Y^4*k-12*Q_Y^3*k+12*k*Q_Y^2-6*k*Q_Y-1+3*Q_Y+k-3*Q_Y^2+2*Q_Y^3)), L_Y = -(27*k*Q_Y^7+9*Q_Y^7-108*Q_Y^6*k-9*Q_Y^6-21*Q_Y^5+189*Q_Y^5*k+60*Q_Y^4-195*Q_Y^4*k-70*Q_Y^3+120*Q_Y^3*k-39*k*Q_Y^2+41*Q_Y^2+5*k*Q_Y-11*Q_Y+1)*Q_Y/(-18*Q_Y^8+72*Q_Y^7-147*Q_Y^6+12*Q_Y^5*k+189*Q_Y^5-160*Q_Y^4-30*Q_Y^4*k+36*Q_Y^3*k+89*Q_Y^3-29*Q_Y^2-24*k*Q_Y^2+8*k*Q_Y+4*Q_Y-k), M_X = k, M_Y = 1, O_X = (-27*Q_Y^10+27*k*Q_Y^10+162*Q_Y^9-135*k*Q_Y^9+324*k*Q_Y^8-459*Q_Y^8+816*Q_Y^7-468*k*Q_Y^7-996*Q_Y^6+420*Q_Y^6*k-210*Q_Y^5*k+858*Q_Y^5+14*Q_Y^4*k-518*Q_Y^4+208*Q_Y^3+56*Q_Y^3*k-37*k*Q_Y^2-49*Q_Y^2+5*Q_Y+10*k*Q_Y-k)/((-18*Q_Y^8+72*Q_Y^7-147*Q_Y^6+12*Q_Y^5*k+189*Q_Y^5-160*Q_Y^4-30*Q_Y^4*k+36*Q_Y^3*k+89*Q_Y^3-29*Q_Y^2-24*k*Q_Y^2+8*k*Q_Y+4*Q_Y-k)*(3*Q_Y^2+1-3*Q_Y)), O_Y = 1, P_X = (48*Q_Y^4*k-44*Q_Y^3*k-30*Q_Y^5*k+9*Q_Y^6*k+25*k*Q_Y^2-8*k*Q_Y-1+5*Q_Y+16*Q_Y^3-15*Q_Y^4-11*Q_Y^2-3*Q_Y^6+9*Q_Y^5+k)/((3*Q_Y^2+1-3*Q_Y)*(6*Q_Y^4*k-12*Q_Y^3*k+12*k*Q_Y^2-6*k*Q_Y-1+3*Q_Y+k-3*Q_Y^2+2*Q_Y^3)), P_Y = (3*Q_Y^4*k-6*Q_Y^3*k+6*k*Q_Y^2-4*k*Q_Y+k+5*Q_Y^2-2*Q_Y-5*Q_Y^3+3*Q_Y^4)/(6*Q_Y^4*k-12*Q_Y^3*k+12*k*Q_Y^2-6*k*Q_Y-1+3*Q_Y+k-3*Q_Y^2+2*Q_Y^3), Q_X = -(1-6*Q_Y+78*Q_Y^5*k-27*Q_Y^3+31*Q_Y^4-39*Q_Y^6*k-3*Q_Y^7+9*k*Q_Y^7+16*Q_Y^2+12*Q_Y^6+9*k*Q_Y-92*Q_Y^4*k-33*k*Q_Y^2+69*Q_Y^3*k-24*Q_Y^5-k)/((3*Q_Y^2+1-3*Q_Y)*(6*Q_Y^4*k-12*Q_Y^3*k+12*k*Q_Y^2-6*k*Q_Y-1+3*Q_Y+k-3*Q_Y^2+2*Q_Y^3)), Q_Y = Q_Y, Y = -(9*Q_Y^4*k+9*k*Q_Y^2-15*Q_Y^3*k+12*Q_Y^3+6*Q_Y-13*Q_Y^2-2*k*Q_Y-3*Q_Y^4-1)/(6*Q_Y^4*k-12*Q_Y^3*k+12*k*Q_Y^2-6*k*Q_Y-1+3*Q_Y+k-3*Q_Y^2+2*Q_Y^3) (4)RootOf(1+2*_Z+2*_Z^2+_Z^4-2*_Z^3) = m, RootOf(_Z^2+1) = n B_Y = -RootOf(-1-_Z*n+_Z^2), D_Y = -n, E_X = -(RootOf(-1-_Z*n+_Z^2)-1)*n, E_Y = -(RootOf(-1-_Z*n+_Z^2)-1)*n, G_X = RootOf(-1-_Z*n+_Z^2), H_X = -(RootOf(-1-_Z*n+_Z^2)-1)*n, H_Y = 1-RootOf(-1-_Z*n+_Z^2)+n, I_X = -1-RootOf(-1-_Z*n+_Z^2)*n+RootOf(-1-_Z*n+_Z^2), I_Y = -(RootOf(-1-_Z*n+_Z^2)-1)*n, K_Y = RootOf(-1-_Z*n+_Z^2), L_X = RootOf(-1-_Z*n+_Z^2), L_Y = -(RootOf(-1-_Z*n+_Z^2)-1)*n, M_X = n, M_Y = 1, O_X = -(RootOf(-1-_Z*n+_Z^2)-1)*n, O_Y = 1, P_X = RootOf(-1-_Z*n+_Z^2), P_Y = -RootOf(-1-_Z*n+_Z^2)*n, Q_X = -(1/2)*RootOf(-1-_Z*n+_Z^2)*(-1+n), Q_Y = 1/2+(1/2)*n (5)(RootOf(1+2 _Z+2 _Z^2+_Z^4-2 _Z^3))=m B_Y = -(1/2)*m^2+m-3/2, D_Y = -2-2*m+2*m^2-m^3, E_X = -(1/2)*m^3+m^2-(1/2)*m, E_Y = 2+(1/2)*m-m^2+(1/2)*m^3, G_X = -(1/2)*m^2+1/2+m, H_X = -(1/2)*m^3+m^2-(1/2)*m, H_Y = 3/2-(5/2)*m^2+3*m+m^3, I_X = -1/2-(1/2)*m-(1/2)*m^3+(1/2)*m^2, I_Y = 2+(1/2)*m-m^2+(1/2)*m^3, K_Y = (1/2)*m^2-m+3/2, L_X = -(1/2)*m^2+1/2+m, L_Y = 2+(1/2)*m-m^2+(1/2)*m^3, M_X = m, M_Y = 1, O_X = -(1/2)*m^3+m^2-(1/2)*m, O_Y = 1, P_X = -(1/2)*m^2+1/2+m, P_Y = -(1/2*(m^2-2*m+3))*m, Q_X = -(1/4)*m^3+(1/4)*m^2+(1/4)*m-1/4, Q_Y = 3/4-(5/4)*m+(3/4)*m^2-(1/4)*m^3 具体的请见附件: 123.pdf (163.06 KB, 下载次数: 0)

2009-12-23 13:45 上传

点击文件名下载附件

mathe

非常好，发现第三个解中M_X=k，所以得出第三个方程也必然没有实数解。证明了17棵树最优整数解是15行

mathe

此外，我们也可以得出17棵树16行的实数坐标范围内的解在射影变换等价意义下唯一，也就是256#给出的解。

wayne

12# mathe 我也来玩玩： 2mathe.zip (119.32 KB, 下载次数: 5)

2009-12-24 09:39 上传

点击文件名下载附件

mathe

你这个里面的图片看不到。 要不然你们再处理一下链接 http://bbs.emath.ac.cn/viewthrea ... fromuid=20#pid24441 中的附件中所有的方程。不过里面大部分很显然只有复数解，就不要管了，主要将所有的有理数解找出来

wayne

忘了把html附带的文件夹给上传了，现已更新。。。

mathe

嗯，你这个结果非常清楚。很显然都没有实数解。

mathe

18棵树18行方程1maxima又求解失败) solve([+1+1*H_Y*Q_X-1*Q_Y,+1*A_X+1*Q_X+1*H_Y*Q_X+1*P_Y*Q_X-2*Q_X*Q_X+1*Q_Y+1*P_Y*Q_Y-4*Q_X*Q_Y-2*Q_Y*Q_Y,+1*D_Y+1*H_Y*Q_X-2*Q_Y-1*H_Y*Q_Y,+1*H_Y-1*H_Y*L_Y+1*H_Y*P_Y-1*L_Y*Q_X+2*P_Y*Q_X-1*Q_X*Q_X+1*Q_Y-1*H_Y*Q_Y-1*L_Y*Q_Y-2*Q_X*Q_Y-1*Q_Y*Q_Y,+1*A_X*H_Y-1*Q_X-1*H_Y*Q_X-1*P_Y*Q_X+1*Q_X*Q_X-2*Q_Y-1*H_Y*Q_Y-1*P_Y*Q_Y+2*Q_X*Q_Y+1*Q_Y*Q_Y,+1*J_Y-1*H_Y*P_Y-1*Q_X+1*H_Y*Q_X-1*P_Y*Q_X+1*Q_X*Q_X-3*Q_Y-1*P_Y*Q_Y+2*Q_X*Q_Y+1*Q_Y*Q_Y,+1*H_Y*J_Y+1*Q_Y+1*H_Y*Q_Y,+1*L_Y-1*Q_X+1*H_Y*Q_X-1*L_Y*Q_X+1*P_Y*Q_X-1*Q_Y-1*L_Y*Q_Y-1*P_Y*Q_Y,+1*A_X*L_Y+1*L_Y*Q_X-1*P_Y*Q_X-1*Q_Y+1*L_Y*Q_Y,+1*D_Y*L_Y-1*J_Y*L_Y+1*H_Y*Q_X-2*Q_Y-1*H_Y*Q_Y,+1*P_Y-1*Q_X-1*P_Y*Q_X+1*Q_X*Q_X-2*Q_Y-1*P_Y*Q_Y+2*Q_X*Q_Y+1*Q_Y*Q_Y,+1*J_Y*P_Y+1*H_Y*Q_X-1*J_Y*Q_X-1*Q_Y-1*J_Y*Q_Y,+1*L_Y*P_Y-1*Q_X-1*L_Y*Q_X+1*Q_X*Q_X-1*Q_Y-1*L_Y*Q_Y-1*P_Y*Q_Y+2*Q_X*Q_Y+1*Q_Y*Q_Y,+1*P_Y*P_Y-1*P_Y*Q_X+1*Q_X*Q_X-1*P_Y*Q_Y+2*Q_X*Q_Y+1*Q_Y*Q_Y,+1*D_Y*Q_X+1*H_Y*Q_X-1*J_Y*Q_X-1*P_Y*Q_X-1*Q_Y+1*D_Y*Q_Y-1*J_Y*Q_Y-1*P_Y*Q_Y,+1*J_X+1*J_Y,+1*K_Y-1*P_Y+1*Q_X+1*Q_Y,-1+1*L_X+1*L_Y,+1*O_X-1*Q_X-1*Q_Y,+1*P_X+1*P_Y-1*Q_X-1*Q_Y,+1*C_X+1*P_Y-1*Q_X-1*Q_Y,-1+1*C_Y,+1*K_X+1*P_Y-1*Q_X-1*Q_Y,+1*D_X+1*J_Y,+1*B_Y-1*P_Y,+1+1*R_Y,+1*E_X+1*J_Y,+1*A_Y-1*P_Y,-1+1*E_Y],[A_X,D_Y,H_Y,J_Y,L_Y,P_Y,Q_X,Q_Y,J_X,K_Y,L_X,O_X,P_X,C_X,C_Y,K_X,D_X,B_Y,R_Y,E_X,A_Y,E_Y]); 解出来了，有点慢，4次方程4个复数根，没有实数解