方程组求解（或无解证明）

mathe

请大家用数学软件验算一下，用maxima解不出来： solve([+1-1*Q_X-1*Q_Y+1*L_X*Q_Y,+1*I_Y+1*P_X-1*I_Y*P_X,+1*J_Y-1*I_Y*Q_X-2*J_Y*Q_X-1*Q_Y+1*L_X*Q_Y+2*O_X*Q_Y,+1*K_X+1*O_Y-1*L_X*O_Y-1*Q_X-1*Q_Y+1*L_X*Q_Y,+1*I_Y*K_X-1*L_X*O_Y-1*Q_X-1*Q_Y+1*L_X*Q_Y,+1*J_Y*K_X-1*K_X*O_Y+1*Q_X+1*Q_Y-1*L_X*Q_Y,+1*L_X-1*I_Y*L_X-1*O_Y-1*P_X+1*I_Y*P_X+1*I_Y*Q_X+2*J_Y*Q_X+1*Q_Y-1*L_X*Q_Y-2*O_X*Q_Y,+1*J_Y*L_X-1*O_Y-1*L_X*O_Y-1*P_X+1*I_Y*P_X,+1*O_X-1*Q_X-1*O_X*Q_Y,+1*I_Y*O_X+1*P_X-1*I_Y*P_X-1*I_Y*Q_X-2*J_Y*Q_X+1*O_X*Q_Y,+1*J_Y*O_X+1*I_Y*P_X-1*Q_X-1*O_X*Q_Y,+1*O_X*O_Y+1*P_X-1*Q_X-1*O_X*Q_Y,+1*J_Y*P_X-1*I_Y*Q_X-2*J_Y*Q_X+1*O_X*Q_Y,+1*O_Y*P_X-1*Q_X-1*O_X*Q_Y,+1*O_Y*Q_X-1*O_X*Q_Y,+1+1*D_Y-1*I_Y,-1+1*H_Y+1*I_Y+1*J_Y-1*O_Y,+1*I_X-1*P_X,+1*J_X-1*O_X,+1*F_Y-1*O_Y,+1+1*A_Y-1*I_Y-1*J_Y+1*O_Y,+1*E_X-1*O_X,+1*K_Y-1*O_Y,-1+1*L_Y,+1*F_X-1*P_X,-1+1*P_Y],[I_Y,J_Y,K_X,L_X,O_X,O_Y,P_X,Q_X,Q_Y,D_Y,H_Y,I_X,J_X,F_Y,A_Y,E_X,K_Y,L_Y,F_X,P_Y]); （应该是无解的） 17.17.1

mathe

第二个: solve([+1-1*F_Y*P_Y+1*P_Y*P_Y-1*F_Y*Q_X-2*G_Y*Q_X+1*P_Y*Q_X-1*P_Y*Q_Y,+1*F_Y-1*F_Y*P_Y-1*G_Y*P_Y+1*P_Y*P_Y,+1*G_Y-1*F_Y*Q_X-2*G_Y*Q_X+1*P_Y*Q_X-1*P_Y*Q_Y,+1*L_X+1*Q_Y-2*L_X*Q_Y-1*M_X*Q_Y,+1*F_Y*L_X-1*G_Y*P_Y,+1*G_Y*L_X+1*G_Y*P_Y-1*L_X*P_Y-1*F_Y*Q_X-2*G_Y*Q_X+1*P_Y*Q_X-1*P_Y*Q_Y,+1*M_X-2*F_Y*P_Y+2*L_X*P_Y+1*M_X*P_Y+1*P_Y*P_Y-2*F_Y*Q_X-4*G_Y*Q_X+2*P_Y*Q_X+1*Q_Y-2*L_X*Q_Y-1*M_X*Q_Y-2*P_Y*Q_Y,+1*F_Y*M_X-1*F_Y*P_Y+1*P_Y*P_Y-1*F_Y*Q_X-2*G_Y*Q_X+1*P_Y*Q_X-1*P_Y*Q_Y,+1*G_Y*M_X+1*F_Y*P_Y-1*M_X*P_Y+1*F_Y*Q_X+2*G_Y*Q_X-1*P_Y*Q_X+1*P_Y*Q_Y,+1*P_Y-1*F_Y*P_Y-1*F_Y*Q_X-2*G_Y*Q_X+1*P_Y*Q_X-1*P_Y*Q_Y,+1*Q_X-1*Q_Y+1*P_Y*Q_Y,-2+1*K_X+1*L_X+1*M_X+1*P_Y,-1+1*P_X+1*P_Y,+1*A_Y+1*G_Y-1*P_Y,-1+1*E_Y+1*F_Y+1*G_Y-1*P_Y,+1*C_Y+1*F_Y+1*G_Y-1*P_Y,+1*J_Y-1*P_Y,+1*M_Y-1*P_Y,-1+1*K_Y,-1+1*J_X,-1+1*I_X+1*P_Y,-1+1*E_X,-1+1*I_Y,-1+1*F_X+1*P_Y],[F_Y,G_Y,L_X,M_X,P_Y,Q_X,Q_Y,K_X,P_X,A_Y,E_Y,C_Y,J_Y,M_Y,K_Y,J_X,I_X,E_X,I_Y,F_X]); 17.16.3

mathe

solve([+1-1*P_Y-1*K_Y*Q_X-1*Q_Y-1*M_X*Q_Y+1*O_X*Q_Y,+1*K_Y-1*L_Y*O_X-1*P_Y+1*O_X*P_Y-1*K_Y*Q_X-1*Q_Y-1*M_X*Q_Y+1*O_X*Q_Y,+1*L_Y+1*K_Y*O_X-1*L_Y*O_X-1*P_Y+1*O_X*P_Y-1*K_Y*Q_X-1*Q_Y-1*M_X*Q_Y+1*O_X*Q_Y,+1*M_X+1*O_X*P_Y+1*P_X*P_Y-2*Q_X+1*K_Y*Q_X+1*Q_Y+1*M_X*Q_Y-1*O_X*Q_Y-2*P_X*Q_Y,+1*K_Y*M_X+1*P_Y,+1*L_Y*M_X+1*P_Y-1*Q_X+1*K_Y*Q_X+1*Q_Y+1*M_X*Q_Y-1*O_X*Q_Y-1*P_X*Q_Y,+1*O_X-1*P_Y+1*P_X*P_Y-1*Q_X-1*P_X*Q_Y,+1*P_X-1*Q_X-1*P_X*Q_Y,+1*K_Y*P_X-1*K_Y*Q_X-1*Q_Y-1*M_X*Q_Y+1*O_X*Q_Y,+1*L_Y*P_X+1*P_Y-1*P_X*P_Y,+1*M_X*P_Y-1*Q_X-1*P_X*Q_Y,+1*L_Y*Q_X+1*Q_Y+1*M_X*Q_Y-1*O_X*Q_Y-1*P_X*Q_Y,+1+1*I_X+1*M_X-1*O_X-1*P_X,+1*D_Y+1*L_Y-1*P_Y,-1+1*H_Y+1*K_Y-1*L_Y+1*P_Y,+1*L_X-1*P_X,+1*E_Y-1*L_Y,-1+1*O_Y,+1*I_Y-1*L_Y,+1*G_X-1*P_X,+1*H_X-1*O_X,-1+1*M_Y,+1*E_X-1*O_X,+1*B_Y+1*K_Y],[K_Y,L_Y,M_X,O_X,P_X,P_Y,Q_X,Q_Y,I_X,D_Y,H_Y,L_X,E_Y,O_Y,I_Y,G_X,H_X,M_Y,E_X,B_Y]); 17.16.6

mathe

solve([+1-1*O_X*P_X+1*P_Y-1*O_X*P_Y-1*Q_Y-1*N_Y*Q_Y+1*O_X*Q_Y-1*P_Y*Q_Y+1*Q_Y*Q_Y,+1*N_Y-1*N_Y*P_X-1*P_Y*Q_Y,+1*N_Y*N_Y-1*O_X*O_X-1*N_Y*P_X+2*O_X*P_X-2*P_Y+2*O_X*P_Y-1*N_Y*Q_Y-1*O_X*Q_Y+1*P_Y*Q_Y,+1*O_X-1*O_X*P_X+1*P_Y-1*O_X*P_Y-1*P_Y*Q_Y,+1*N_Y*O_X-1*O_X*O_X-1*N_Y*P_X+1*O_X*P_X-1*P_Y+1*O_X*P_Y-1*O_X*Q_Y,+1*P_X-1*O_X*P_X+1*P_Y-1*P_X*Q_Y-1*P_Y*Q_Y,+1*N_Y*P_Y-1*P_X*Q_Y-1*P_Y*Q_Y,+1*D_Y-1*N_Y+1*O_X,+1+1*L_Y-1*O_X-1*Q_Y,+1*A_Y-1*N_Y+1*O_X+1*Q_Y,+1*B_X+1*N_Y-1*O_X,+1*G_X-1*N_Y+1*Q_Y,-1+1*Q_X+1*Q_Y,+1*F_X-1*O_X,+1*D_X-1*O_X,+1*M_Y-1*N_Y,+1*G_Y-1*Q_Y,+1*F_Y-1*Q_Y,+1+1*H_Y,+1*B_Y-1*N_Y,-1+1*M_X+1*Q_Y,-1+1*L_X+1*Q_Y],[N_Y,O_X,P_X,P_Y,Q_Y,D_Y,L_Y,A_Y,B_X,G_X,Q_X,F_X,D_X,M_Y,G_Y,F_Y,H_Y,B_Y,M_X,L_X]); 17.16.7

mathe

solve([+1-1*M_X*P_X+1*N_Y*P_X-1*P_Y-1*N_X*P_Y,+1*M_X-1*Q_X-1*M_X*Q_X+1*N_Y*Q_X,+1*M_X*M_X-2*M_X*N_Y-2*M_X*P_X+3*N_Y*P_X-1*P_Y-2*N_X*P_Y+2*Q_X+2*M_X*Q_X-2*N_Y*Q_X,+1*N_X-2*M_X*P_X+3*N_Y*P_X-1*P_Y-2*N_X*P_Y+1*Q_X+1*M_X*Q_X-2*N_Y*Q_X,+1*M_X*N_X+1*M_X*P_X-1*N_Y*P_X+1*P_Y+1*N_X*P_Y-2*Q_X-2*M_X*Q_X+3*N_Y*Q_X,+1*N_Y+1*M_X*P_X-2*N_Y*P_X+1*N_X*P_Y-1*Q_X-1*M_X*Q_X+1*N_Y*Q_X,+1*N_X*N_Y-1*Q_X-1*M_X*Q_X+1*N_Y*Q_X,+1*P_X-1*N_X*P_Y,-1+1*K_X+1*N_X,+1+1*L_Y+1*M_X-2*N_Y,+1+1*F_Y+1*M_X-1*N_Y,-1+1*G_Y-1*N_Y,+1+1*A_Y+1*M_X-2*N_Y,+1*M_Y-1*N_Y,+1*G_X-1*N_X,-1+1*E_X,+1*E_Y-1*N_Y,+1*J_X-1*N_X,-1+1*J_Y,-1+1*Q_Y,-1+1*C_Y-1*M_X+1*N_Y,-1+1*L_X],[M_X,N_X,N_Y,P_X,P_Y,Q_X,K_X,L_Y,F_Y,G_Y,A_Y,M_Y,G_X,E_X,E_Y,J_X,J_Y,Q_Y,C_Y,L_X]); 17.16.9

数学星空

呵,结果请见附件: mathe.pdf (290.71 KB, 下载次数: 9)

2009-12-23 12:05 上传

点击文件名下载附件

mathe

非常不错， 也就是第一个方程的解是Q_Y=RootOf(3z^2-3z+1) 第二个方程的解是J_Y=RootOf(z^2-z+1) (但是此后A_Y还有多种取值) 第三个方程的前两个解是B_Y=-RootOf(z^2-z+1)，但是第三个方程比较有意思，请问将Q_Y代入第三个方程是不是结果全部是有理数？请数学星空化简一下

mathe

我需要验证第一个方程没有实数解（已经得到验证），还有后面4个方程都没有有理数解，但是现在不确定。

mathe

发现第三个方程只有第三个解比较麻烦，但是其中包含 rootOf((3Q_Y^2+1-3Q_Y)_Z^2+(-1+2Q_Y)_Z+1-Q_Y+Q_Y^2) 这个没有实数解，所以应该能够证明第三个解不是实数解的，只是有些复杂。 能不能将第三个解中这个rootOf用一个参数t替换然后再显示出来，此外，含有上面参数的最简单的表达式是哪个变量？

数学星空

"第三个方程的前两个解是B_Y=-RootOf(z^2-z+1)，但是第三个方程比较有意思，请问将Q_Y代入第三个方程是不是结果全部是有理数？请数学星空化简一下 " 不明白,你说的是"Q_Y=??"代入第三个方程(f3),求解是否有有理数解????