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[提问] 方程组求解(或无解证明)

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发表于 2009-12-23 11:26:16 | 显示全部楼层 |阅读模式

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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
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2009-12-23 11:30:03 | 显示全部楼层
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
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2009-12-23 11:31:26 | 显示全部楼层
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
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2009-12-23 11:32:12 | 显示全部楼层
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
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2009-12-23 11:33:02 | 显示全部楼层
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
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2009-12-23 12:05:36 | 显示全部楼层
呵,结果请见附件:
mathe.pdf (290.71 KB, 下载次数: 9)
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2009-12-23 12:21:45 | 显示全部楼层
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
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2009-12-23 12:26:48 | 显示全部楼层
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。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2009-12-23 12:39:08 | 显示全部楼层
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?
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2009-12-23 12:41:14 | 显示全部楼层
本帖最后由 数学星空 于 2009-12-23 12:49 编辑

"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????
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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