消元难题

数学星空

由于我们遇到的许多难题几乎都可以转化为一个方程组的消元问题，因此我们有必要利用数学软件来解决我们所遇到的难题： $f(x_1,x_2)=m_1 .....................(1)$ $f(x_2,x_3)=m_2 .....................(2)$ ...........................……........................ $f(x_(k-1),x_k)=m_(k-1) ...........(k-1)$ $f(x_k,x_1)=m_k.......................(k)$ 其中$x_i,(1<=i<=k)$是变量，$m_l,(1<=l<=k)$是常量 消元$x_2,x_3,...,x_k $,求得：$g(x_1)=0$ 请大家给出好的求解的方案？

数学星空

例如：http://bbs.emath.ac.cn/viewthread.php?tid=4289&extra=&page=2 对于$N=5$,我们可以得到：（$m,n,a$均为常量，$x_1,x_2,x_3,x_4,x_5$为变量满足下面方程） $(-1-6*y^2-2*x^2+4*y-y^4-2*y^2*x^2+4*y^3-x^4+4*x^2*y)*a^2+4*x^2*n^2*y^2-8*x^2*n^2*y+4*x^2*n^2+4*x^4*m^2-16*n^2*y^3+32*n^2*y^2-16*n^2*y-16*y*x^2*m^2$＝0 其中： $(x,y)=(x_1+x_2,x_1*x_2)=(x_2+x_3,x_2*x_3)=(x_3+x_4,x_3*x_4)=(x_4+x_5,x_4*x_5)=(x_5+x_1,x_1*x_5)$ 如何消元得到$f(x_1)=0$ (注：直接消元是不可能的）

mathe

可以试一试多变量的牛顿迭代法是否可行。 可以任意选择初始值向量 $X=[(x_1),(x_2),(...),(x_n)]$ 我们记$f_k$表示$f(x_k,x_{k+1})$,而$f_{1k}$表示$f(x_k,x_{k+1})$的第一个偏导数,$f_{2k}$表示第二个偏导数 计算一阶偏导数矩阵 $A=[(f_{11},f_{21},0,...,0),(0,f_{12},f_{22},...,0),(...,...,...,...,...),(f_{2n},0,0,...,f_{1n})]$ $b=[(f_1-m_1),(f_2-m_2),(...),(f_n-m_n)]$ 然后迭代计算 $X=X-A^-1b$ 如果多次迭代没有收敛趋势，可以试着另外选择初始点看看

数学星空

其实，我需要一个符号计算方案，而不是数值计算哈。。。 由于代数计算过程中计算量急剧膨胀（呈指数），如何寻找计算不变量来简化计算过程？ 当然若能将方程参数表示，再使用迭代计算也是一种好方案。 但这些想法似乎都很难实现！

mathe

这个具体问题具体分析，同f性质明显相关

数学星空

对于椭圆：$x^2/5^2+y^2/3^2=1$，内接等五边形（凸形）的边长为$a$ 可以令$x_k=(5*(alpha_k+1/alpha_k))/2,y=(3*(alpha_k-1/alpha_k))/(2*i),i^2=-1$ 则$(x_1-x_2)^2+(y_1-y_2)^2=a^2$ 代入得：$alpha_1^2*(25*(-alpha_2+1)^2-9*(1+alpha_2)^2)=4*a^2$ 可以参数有理化： $alpha_1=-(17/2)*(a*(32*t*a^2+8*a^2-32*t*a-16*a-225*t)*t)/(16*a^2-16*a^3+4*a^4+450*t*a-161*t*a^2-96*t*a^3+32*t*a^4-900*t^2+900*t^2*a-836*t^2*a^2-128*t^2*a^3+64*t^2*a^4)$ $alpha_2=-(1/68)*(256*t^2*a^4+128*t*a^4+16*a^4-512*t^2*a^3-384*t*a^3-64*a^3-7968*t^2*a^2-1800*t*a^2+64*a^2+8224*t^2*a+4112*t*a+61425*t^2)/((32*t*a^2+8*a^2-32*t*a-16*a-225*t)*t)$ 即$(x_1-x_2,1/(x_1*x_2))=(x_2-x_3,1/(x_2*x_3))=(x_3-x_4,1/(x_3*x_4))=(x_4-x_5,1/(x_4*x_5))=(x_5-x_1,1/(x_1*x_5))=(alpha_1,alpha_2)$下一步如何消去$x_2,x_3,x_4,x_5?$

数学星空

下面问题已经困扰了我好长时间了：

试给出x关于{a,b,c}的代数方程？

\(  4(x+y+c)r^2-(x+y-c)(-x+y+c)(x-y+c)＝0 \)
\( 4(x+z+b)r^2-(x+z-b)(-x+z+b)(x-z+b)＝0 \)
\( 4(y+z+a)r^2-(y+z-a)(-y+z+a)(y-z+a)＝0 \)
\( 4r^2(2x+2y+2z+a+b+c)^2+a^4-2a^2b^2-2a^2c^2+b^4-2b^2c^2+c^4＝0 \)

通过多种方法求解：我只能得到下面几个结果，但仍未解决 x关于{a,b,c}的代数方程？

(1)\(r\)关于{a,b,c}的代数方程：

1. (a+b+c)^4(-c+a+b)^6(-c+a-b)^6(c+a-b)^6+8(9a^2+10ab+10ac+9b^2+10bc+9c^2)(a+b+c)^3(-c+a+b)^5(-c+a-b)^5(c+a-b)^5r^2+
   
2. 16(125a^4+268a^3b+268a^3c+382a^2b^2+580a^2bc+382a^2c^2+268ab^3+580ab^2c+580abc^2+268ac^3+125b^4+268b^3c+382b^2c^2+
   
3. 268bc^3+125c^4)(a+b+c)^2(-c+a+b)^4(-c+a-b)^4(c+a-b)^4r^4+1024(a+b+c)(24a^6+73a^5b+73a^5c+152a^4b^2+248a^4bc+152a^4c^2+
   
4. 190a^3b^3+447a^3b^2c+447a^3bc^2+190a^3c^3+152a^2b^4+447a^2b^3c+630a^2b^2c^2+447a^2bc^3+152a^2c^4+73ab^5+
   
5. 248ab^4c+447ab^3c^2+447ab^2c^3+248abc^4+73ac^5+24b^6+73b^5c+152b^4c^2+190b^3c^3+152b^2c^4+73bc^5+24c^6)(-c+a+b)^3
   
6. (-c+a-b)^3(c+a-b)^3r^6+1024(101a^8+374a^7b+374a^7c+1244a^6b^2+2018a^6bc+1244a^6c^2+2426a^5b^3+5814a^5b^2c+
   
7. 5814a^5bc^2+2426a^5c^3+2974a^4b^4+9202a^4b^3c+13188a^4b^2c^2+9202a^4bc^3+2974a^4c^4+2426a^3b^5+9202a^3b^4c+
   
8. 17012a^3b^3c^2+17012a^3b^2c^3+9202a^3bc^4+2426a^3c^5+1244a^2b^6+5814a^2b^5c+13188a^2b^4c^2+17012a^2b^3c^3+
   
9. 13188a^2b^2c^4+5814a^2bc^5+1244a^2c^6+374ab^7+2018ab^6c+5814ab^5c^2+9202ab^4c^3+9202ab^3c^4+5814ab^2c^5+
   
10. 2018abc^6+374ac^7+101b^8+374b^7c+1244b^6c^2+2426b^5c^3+2974b^4c^4+2426b^3c^5+1244b^2c^6+374bc^7+101c^8)(-c+a+b)^2
    
11. (-c+a-b)^2(c+a-b)^2r^8-16384(-c+a+b)(c+a-b)(-c+a-b)(6a^9+33a^8b+33a^8c-129a^7b^2+24a^7bc-129a^7c^2-383a^6b^3-474a^6b^2c-
    
12. 474a^6bc^2-383a^6c^3-551a^5b^4-1808a^5b^3c-1944a^5b^2c^2-1808a^5bc^3-551a^5c^4-551a^4b^5-2718a^4b^4c-4109a^4b^3c^2-
    
13. 4109a^4b^2c^3-2718a^4bc^4-551a^4c^5-383a^3b^6-1808a^3b^5c-4109a^3b^4c^2-4920a^3b^3c^3-4109a^3b^2c^4-1808a^3bc^5-
    
14. 383a^3c^6-129a^2b^7-474a^2b^6c-1944a^2b^5c^2-4109a^2b^4c^3-4109a^2b^3c^4-1944a^2b^2c^5-474a^2bc^6-129a^2c^7+33ab^8+
    
15. 24ab^7c-474ab^6c^2-1808ab^5c^3-2718ab^4c^4-1808ab^3c^5-474ab^2c^6+24abc^7+33ac^8+6b^9+33b^8c-129b^7c^2-383b^6c^3-
    
16. 551b^5c^4-551b^4c^5-383b^3c^6-129b^2c^7+33bc^8+6c^9)r^{10}+(262144a^{10}+262144a^9b+262144a^9c-3866624a^8b^2-262144a^8bc-
    
17. 3866624a^8c^2-20054016a^7b^3-9306112a^7b^2c-9306112a^7bc^2-20054016a^7c^3+3604480a^6b^4-2490368a^6b^3c+65732608a^6b^2c^2-
    
18. 2490368a^6bc^3+3604480a^6c^4+39583744a^5b^5+11796480a^5b^4c+91881472a^5b^3c^2+91881472a^5b^2c^3+11796480a^5bc^4+
    
19. 39583744a^5c^5+3604480a^4b^6+11796480a^4b^5c+80216064a^4b^4c^2+362676224a^4b^3c^3+80216064a^4b^2c^4+11796480a^4bc^5+
    
20. 3604480a^4c^6-20054016a^3b^7-2490368a^3b^6c+91881472a^3b^5c^2+362676224a^3b^4c^3+362676224a^3b^3c^4+91881472a^3b^2c^5-
    
21. 2490368a^3bc^6-20054016a^3c^7-3866624a^2b^8-9306112a^2b^7c+65732608a^2b^6c^2+91881472a^2b^5c^3+80216064a^2b^4c^4+
    
22. 91881472a^2b^3c^5+65732608a^2b^2c^6-9306112a^2bc^7-3866624a^2c^8+262144ab^9-262144ab^8c-9306112ab^7c^2-2490368ab^6c^3+
    
23. 11796480ab^5c^4+11796480ab^4c^5-2490368ab^3c^6-9306112ab^2c^7-262144abc^8+262144ac^9+262144b^{10}+262144b^9c-3866624b^8c^2-
    
24. 20054016b^7c^3+3604480b^6c^4+39583744b^5c^5+3604480b^4c^6-20054016b^3c^7-3866624b^2c^8+262144bc^9+
    
25. 262144c^{10})r^{12}+(9437184a^6b^2+9437184a^6c^2+9437184a^5b^2c+9437184a^5bc^2-18874368a^4b^4-9437184a^4b^3c-
    
26. 113246208a^4b^2c^2-9437184a^4bc^3-18874368a^4c^4-9437184a^3b^4c-452984832a^3b^3c^2-452984832a^3b^2c^3-9437184a^3bc^4+
    
27. 9437184a^2b^6+9437184a^2b^5c-113246208a^2b^4c^2-452984832a^2b^3c^3-113246208a^2b^2c^4+9437184a^2bc^5+9437184a^2c^6+
    
28. 9437184ab^5c^2-9437184ab^4c^3-9437184ab^3c^4+9437184ab^2c^5+9437184b^6c^2-18874368b^4c^4+9437184b^2c^6)r^{14}+339738624a^2b^2c^2r^{16}＝0

复制代码

（2） \(x+y+z=L\)关于{a,b,c}的代数方程

\((c+a+b)^2(a^3-a^2b-a^2c-ab^2+6abc-ac^2+b^3-b^2c-bc^2+c^3)^2-(c+a+b)(11a^5-5a^4b-5a^4c+10a^3b^2+8a^3bc+10a^3c^2+10a^2b^3+10a^2b^2c+10a^2bc^2+10a^2c^3-5ab^4+8ab^3c+10ab^2c^2+8abc^3-5ac^4+11b^5-5b^4c+10b^3c^2+10b^2c^3-5bc^4+11c^5)L^2+(44a^4+32a^3b+32a^3c+40a^2b^2+32a^2bc+40a^2c^2+32ab^3+32ab^2c+32abc^2+32ac^3+44b^4+32b^3c+40b^2c^2+32bc^3+44c^4)L^4+(-80a^2-32ab-32ac-80b^2-32bc-80c^2)L^6+64L^8=0\)

1. 2085*a^3*b^4*c*x^4-424*a^3*b^3*c^5*x-736*a^3*b^3*c^4*x^2+3520*a^3*b^3*c^3*x^3-1206*a^3*b^3*c^2*x^4-4124*a^3*b^3*c*x^5-276*a^3*b^2*c^6*x+144*a^3*b^2*c^5*x^2+2080*a^3*b^2*c^4*x^3-
   
2. 1206*a^3*b^2*c^3*x^4-7474*a^3*b^2*c^2*x^5-4038*a^3*b^2*c*x^6-24*a^3*b*c^7*x-240*a^3*b*c^6*x^2+1248*a^3*b*c^5*x^3+2085*a^3*b*c^4*x^4-4124*a^3*b*c^3*x^5-4038*a^3*b*c^2*x^6+396*a^3*b*c*x^7-
   
3. 19*a^2*b^8*c*x-56*a^9*b*c*x-36*a^8*b^2*c*x-36*a^8*b*c^2*x-168*a^8*b*c*x^2-486*a*b^5*c^4*x^2+168*a*b^5*c^3*x^3-534*a*b^5*c^2*x^4-2164*a*b^5*c*x^5-486*a*b^4*c^5*x^2-214*a*b^4*c^4*x^3+
   
4. 550*a*b^4*c^3*x^4-850*a*b^4*c^2*x^5-633*a*b^4*c*x^6+44*a*b^3*c^6*x^2+168*a*b^3*c^5*x^3+550*a*b^3*c^4*x^4-344*a*b^3*c^3*x^5+2766*a*b^3*c^2*x^6-84*a^8*c^2*x^2+383*a^5*b^5*x^2+
   
5. 1107*a^4*b*x^7+89*a^7*b^4*x+183*a^5*b*x^6-46*a^7*b^3*x^2-12*a^9*b^2*c-1012*a^5*c^3*x^4-1424*a^4*b^3*x^5+531*a^4*b^5*x^3-84*a^8*b^2*x^2-46*a^5*b^6*c-42*a^5*b^5*c^2+50*a^7*b^3*c^2+
   
6. 69*a^6*c^5*x-180*a^8*b*x^3-12*a^8*c^3*x-70*a^4*c^7*x-292*a^5*b^2*x^5-279*a^6*c*x^5-246*a^5*b^4*c^3-42*a^5*b^2*c^5-28*a^9*b^2*x-34*a^5*b^6*x-1012*a^5*b^3*x^4-846*a^6*c^2*x^4+92*b^2*c^7*x^3+
   
7. 532*b^2*c^6*x^4-394*b^2*c^5*x^5-2310*b^2*c^4*x^6+1494*b^2*c^3*x^7+5922*b^2*c^2*x^8+972*b^2*c*x^9+15*b*c^8*x^3+264*b*c^7*x^4+594*b*c^6*x^5-2076*b*c^5*x^6-3213*b*c^4*x^7+5004*b*c^3*x^8+
   
8. 972*b*c^2*x^9-8424*b*c*x^10-972*a^2*c*x^9-21*a*b^9*x^2+183*a^5*c*x^6-382*a^7*c^2*x^3-159*a^7*c*x^4+69*a^6*b^5*x+209*a^6*b^4*x^2+1107*a^4*c*x^7-15*a^3*b^8*c-35*a^3*b^8*x+
   
9. 52*a^3*b^7*c^2-192*a^3*b^7*x^2+196*a^3*b^6*c^3+416*a^3*b^6*x^3-2*a^3*b^5*c^4+417*a^3*b^5*x^4-2*a^3*b^4*c^5-1283*a^3*b^4*x^5+196*a^3*b^3*c^6+414*a^3*b^3*x^6+52*a^3*b^2*c^7-18*a^3*b^2*x^7-
   
10. 15*a^3*b*c^8-936*a^3*b*x^8-35*a^3*c^8*x-192*a^3*c^7*x^2+416*a^3*c^6*x^3+417*a^3*c^5*x^4-1283*a^3*c^4*x^5+414*a^3*c^3*x^6-18*a^3*c^2*x^7-936*a^3*c*x^8+5*a^2*b^9*x-31*a^2*b^8*x^2+
    
11. 24*a^2*b^7*x^3+39*a*b^8*x^3+162*a*b^7*x^4-462*a*b^6*x^5-357*a*b^5*x^6+1431*a*b^4*x^7-72*a*b^3*x^8+1296*a*b*x^10-21*a*c^9*x^2+39*a*c^8*x^3+162*a*c^7*x^4-462*a*c^6*x^5-
    
12. 357*a*c^5*x^6+1431*a*c^4*x^7-72*a*c^3*x^8+1296*a*c*x^10+15*b^8*c*x^3+92*b^7*c^2*x^3+264*b^7*c*x^4-20*b^6*c^3*x^3+532*b^6*c^2*x^4+594*b^6*c*x^5-334*b^5*c^4*x^3+184*b^5*c^3*x^4-
    
13. 394*b^5*c^2*x^5-2076*b^5*c*x^6-334*b^4*c^5*x^3-222*b^4*c^4*x^4-30*a^4*c^6*x^2+531*a^4*c^5*x^3+135*a^4*c^4*x^4-1424*a^4*c^3*x^5+264*a^4*c^2*x^6+826*b^4*c^3*x^5-2310*b^4*c^2*x^6-
    
14. 3213*b^4*c*x^7-20*b^3*c^6*x^3+184*b^3*c^5*x^4+826*b^3*c^4*x^5-648*b^3*c^3*x^6+1494*b^3*c^2*x^7+5004*b^3*c*x^8-12*a^9*b*c^2-94*a^6*b^3*x^3-34*a^5*c^6*x-846*a^6*b^2*x^4-246*a^5*b^3*c^4-
    
15. 30*a^4*b^6*x^2-12*a^8*b^3*x+383*a^5*c^5*x^2-279*a^6*b*x^5-70*a^4*b^7*x-28*a^9*c^2*x+235*a^5*b^4*x^3-292*a^5*c^2*x^5+235*a^5*c^4*x^3-46*a^5*b*c^6+135*a^4*b^4*x^4-94*a^6*c^3*x^3-
    
16. 60*a^9*b*x^2+209*a^6*c^4*x^2+65*a^7*b^4*c-60*a^9*c*x^2+200*a^2*b^6*x^4-343*a^2*b^5*x^5-267*a^2*b^4*x^6+630*a^2*b^3*x^7-234*a^2*b^2*x^8-972*a^2*b*x^9+5*a^2*c^9*x-31*a^2*c^8*x^2+
    
17. 24*a^2*c^7*x^3+200*a^2*c^6*x^4-343*a^2*c^5*x^5-267*a^2*c^4*x^6+630*a^2*c^3*x^7-234*a^2*c^2*x^8+50*a^7*b^2*c^3-382*a^7*b^2*x^3+65*a^7*b*c^4-159*a^7*b*x^4+89*a^7*c^4*x-46*a^7*c^3*x^2-
    
18. 158*a^4*b^2*c^5*x+1022*a^4*b^2*c^4*x^2+846*a^4*b^2*c^3*x^3-5110*a^4*b^2*c^2*x^4-5440*a^4*b^2*c*x^5-74*a^4*b*c^6*x+508*a^4*b*c^5*x^2+2191*a^4*b*c^4*x^3-2164*a^4*b*c^3*x^4-5440*a^4*b*c^2*x^5-
    
19. 1488*a^4*b*c*x^6+3852*a*b^3*c*x^7+188*a*b^2*c^7*x^2+516*a*b^2*c^6*x^3-534*a*b^2*c^5*x^4-850*a*b^2*c^4*x^5+2766*a*b^2*c^3*x^6-1494*a*b^2*c^2*x^7-3528*a*b^2*c*x^8+19*a*b*c^8*x^2+
    
20. 280*a*b*c^7*x^3+206*a*b*c^6*x^4-2164*a*b*c^5*x^5-633*a*b*c^4*x^6+3852*a*b*c^3*x^7-3528*a*b*c^2*x^8-3888*a*b*c*x^9+180*a^2*b^7*c^2*x-8*a^2*b^7*c*x^2+292*a^2*b^6*c^3*x-260*a^2*b^6*c^2*x^2-
    
21. 440*a^2*b^6*c*x^3-202*a^2*b^5*c^4*x-440*a^2*b^5*c^3*x^2-40*a^2*b^5*c^2*x^3+800*a^2*b^5*c*x^4-202*a^2*b^4*c^5*x-314*a^2*b^4*c^4*x^2-824*a^2*b^4*c^3*x^3+2744*a^2*b^4*c^2*x^4+
    
22. 2429*a^2*b^4*c*x^5+292*a^2*b^3*c^6*x-440*a^2*b^3*c^5*x^2-824*a^2*b^3*c^4*x^3+3264*a^2*b^3*c^3*x^4+170*a^2*b^3*c^2*x^5-4236*a^2*b^3*c*x^6-24*a^3*b^7*c*x-276*a^3*b^6*c^2*x-240*a^3*b^6*c*x^2-
    
23. 424*a^3*b^5*c^3*x+144*a^3*b^5*c^2*x^2+1248*a^3*b^5*c*x^3-274*a^3*b^4*c^4*x-736*a^3*b^4*c^3*x^2+2080*a^3*b^4*c^2*x^3+169*a^6*b^4*c*x+162*a^6*b^3*c^2*x+116*a^6*b^3*c*x^2+162*a^6*b^2*c^3*x-
    
24. 186*a^6*b^2*c^2*x^2-1578*a^6*b^2*c*x^3+169*a^6*b*c^4*x+116*a^6*b*c^3*x^2-1578*a^6*b*c^2*x^3-2188*a^6*b*c*x^4+148*a^5*b^5*c*x+386*a^5*b^4*c^2*x+1003*a^5*b^4*c*x^2+408*a^5*b^3*c^3*x+
    
25. 326*a^5*b^3*c^2*x^2-1060*a^5*b^3*c*x^3+386*a^5*b^2*c^4*x+326*a^5*b^2*c^3*x^2-1310*a^5*b^2*c^2*x^3-2876*a^5*b^2*c*x^4+148*a^5*b*c^5*x+1003*a^5*b*c^4*x^2-1060*a^5*b*c^3*x^3-2876*a^5*b*c^2*x^4-
    
26. 2200*a^5*b*c*x^5-74*a^4*b^6*c*x-158*a^4*b^5*c^2*x+508*a^4*b^5*c*x^2-594*a^4*b^4*c^3*x+180*a^2*b^2*c^7*x-260*a^2*b^2*c^6*x^2-40*a^2*b^2*c^5*x^3+2744*a^2*b^2*c^4*x^4+170*a^2*b^2*c^3*x^5-
    
27. 10242*a^2*b^2*c^2*x^6-3438*a^2*b^2*c*x^7-19*a^2*b*c^8*x-8*a^2*b*c^7*x^2-440*a^2*b*c^6*x^3+800*a^2*b*c^5*x^4+2429*a^2*b*c^4*x^5-4236*a^2*b*c^3*x^6-3438*a^2*b*c^2*x^7+5868*a^2*b*c*x^8+
    
28. 19*a*b^8*c*x^2+188*a*b^7*c^2*x^2+280*a*b^7*c*x^3+44*a*b^6*c^3*x^2+516*a*b^6*c^2*x^3+206*a*b^6*c*x^4+1022*a^4*b^4*c^2*x^2+2191*a^4*b^4*c*x^3-594*a^4*b^3*c^4*x+968*a^4*b^3*c^3*x^2+
    
29. 846*a^4*b^3*c^2*x^3-2164*a^4*b^3*c*x^4+52*a^7*b^3*c*x-74*a^7*b^2*c^2*x-506*a^7*b^2*c*x^2+52*a^7*b*c^3*x-506*a^7*b*c^2*x^2-652*a^7*b*c*x^3+264*a^4*b^2*x^6-180*a^8*c*x^3+126*b^7*x^5-
    
30. 225*a^4*x^8+27*b^8*x^4-549*b^4*x^8+25*a^3*b^9-50*a^5*b^7-4*a^9*c^3-90*b^6*x^6+648*a^3*x^9+1620*b^2*x^10-9*c^9*x^3-324*a^2*x^10-108*a^8*x^4-9*b^9*x^3-36*a^9*x^3+29*a^7*b^5+1620*c^2*x^10-729*b^5*x^7-50*a^5*c^7+25*a^3*c^9-4*a^9*b^3+126*c^7*x^5+333*a^6*x^6+27*c^8*x^4+29*a^7*c^5+117*a^7*x^5-90*c^6*x^6+1620*c^3*x^9+1620*b^3*x^9-549*c^4*x^8-405*a^5*x^7-729*c^5*x^7+(4*a^2-8*a*b-8*a*c-16*a*x+4*b^2+8*b*c+16*b*x+4*c^2+16*c*x+16*x^2)*L^10+(8*a^3-16*a^2*b-16*a^2*c-72*a^2*x+16*a*b^2+32*a*b*c+152*a*b*x+16*a*c^2+152*a*c*x+224*a*x^2-8*b^3-
    
31. 24*b^2*c-96*b^2*x-24*b*c^2-192*b*c*x-272*b*x^2-8*c^3-96*c^2*x-272*c*x^2-224*x^3)*L^9+(17*a^4-44*a^3*b-44*a^3*c-160*a^3*x+6*a^2*b^2+108*a^2*b*c+316*a^2*b*x+6*a^2*c^2+
    
32. 316*a^2*c*x+588*a^2*x^2+44*a*b^3-28*a*b^2*c-168*a*b^2*x-28*a*b*c^2-624*a*b*c*x-1200*a*b*x^2+44*a*c^3-168*a*c^2*x-1200*a*c*x^2-1264*a*x^3-23*b^4-12*b^3*c+60*b^3*x+22*b^2*c^2+436*b^2*c*x+
    
33. 868*b^2*x^2-12*b*c^3+436*b*c^2*x+1944*b*c*x^2+1936*b*x^3-23*c^4+60*c^3*x+868*c^2*x^2+1936*c*x^3+1248*x^4)*L^8+(-5*a^5+33*a^4*b+33*a^4*c-70*a^4*x-46*a^3*b^2-76*a^3*b*c+292*a^3*b*x-46*a^3*c^2+
    
34. 292*a^3*c*x+824*a^3*x^2+6*a^2*b^3+50*a^2*b^2*c-88*a^2*b^2*x+50*a^2*b*c^2-1280*a^2*b*c*x-2200*a^2*b*x^2+6*a^2*c^3-88*a^2*c^2*x-2200*a^2*c*x^2-2480*a^2*x^3-13*a*b^4-84*a*b^3*c-452*a*b^3*x-
    
35. 142*a*b^2*c^2+340*a*b^2*c*x+760*a*b^2*x^2-84*a*b*c^3+340*a*b*c^2*x+4896*a*b*c*x^2+5104*a*b*x^3-13*a*c^4-452*a*c^3*x+760*a*c^2*x^2+5104*a*c*x^3+3632*a*x^4+25*b^5+77*b^4*c+318*b^4*x+
    
36. 106*b^3*c^2+200*b^3*c*x-152*b^3*x^2+106*b^2*c^3-236*b^2*c^2*x-3480*b^2*c*x^2-4160*b^2*x^3+77*b*c^4+200*b*c^3*x-3480*b*c^2*x^2-10848*b*c*x^3-7408*b*x^4+25*c^5+318*c^4*x-152*c^3*x^2-
    
37. 4160*c^2*x^3-7408*c*x^4-3424*x^5)*L^7+(-29*a^6+27*a^5*b+27*a^5*c+89*a^5*x-4*a^4*b^2+8*a^4*b*c-211*a^4*b*x-4*a^4*c^2-211*a^4*c*x+14*a^4*x^2+82*a^3*b^3+166*a^3*b^2*c+
    
38. 490*a^3*b^2*x+166*a^3*b*c^2+1044*a^3*b*c*x-388*a^3*b*x^2+82*a^3*c^3+490*a^3*c^2*x-388*a^3*c*x^2-1464*a^3*x^3-49*a^2*b^4-340*a^2*b^3*c-374*a^2*b^3*x-6*a^2*b^2*c^2-610*a^2*b^2*c*x+
    
39. 616*a^2*b^2*x^2-340*a^2*b*c^3-610*a^2*b*c^2*x+5552*a^2*b*c*x^2+7304*a^2*b*x^3-49*a^2*c^4-374*a^2*c^3*x+616*a^2*c^2*x^2+7304*a^2*c*x^3+5320*a^2*x^4-61*a*b^5+79*a*b^4*c+125*a*b^4*x+
    
40. 30*a*b^3*c^2+1348*a*b^3*c*x+2100*a*b^3*x^2+30*a*b^2*c^3+526*a*b^2*c^2*x-2292*a*b^2*c*x^2-2280*a*b^2*x^3+79*a*b*c^4+1348*a*b*c^3*x-2292*a*b*c^2*x^2-19904*a*b*c*x^3-12656*a*b*x^4-
    
41. 61*a*c^5+125*a*c^4*x+2100*a*c^3*x^2-2280*a*c^2*x^3-12656*a*c*x^4-5392*a*x^5+34*b^6-4*b^5*c-247*b^5*x-66*b^4*c^2-1059*b^4*c*x-1798*b^4*x^2-56*b^3*c^3-1174*b^3*c^2*x-1448*b^3*c*x^2+
    
42. 216*b^3*x^3-66*b^2*c^4-1174*b^2*c^3*x+2300*b^2*c^2*x^2+16120*b^2*c*x^3+11880*b^2*x^4-4*b*c^5-1059*b*c^4*x-1448*b*c^3*x^2+16120*b*c^2*x^3+36336*b*c*x^4+16112*b*x^5+34*c^6-247*c^5*x-
    
43. 1798*c^4*x^2+216*c^3*x^3+11880*c^2*x^4+16112*c*x^5+4352*x^6)*L^6+(5*a^7-17*a^6*b-17*a^6*c+140*a^6*x+16*a^5*b^2-128*a^5*b*c-350*a^5*b*x+16*a^5*c^2-350*a^5*c*x-525*a^5*x^2-
    
44. 64*a^4*b^3-144*a^4*b^2*c-76*a^4*b^2*x-144*a^4*b*c^2-72*a^4*b*c*x+461*a^4*b*x^2-64*a^4*c^3-76*a^4*c^2*x+461*a^4*c*x^2+18*a^4*x^3+73*a^3*b^4+484*a^3*b^3*c-64*a^3*b^3*x+118*a^3*b^2*c^2-
    
45. 928*a^3*b^2*c*x-1934*a^3*b^2*x^2+484*a^3*b*c^3-928*a^3*b*c^2*x-4764*a^3*b*c*x^2-844*a^3*b*x^3+73*a^3*c^4-64*a^3*c^3*x-1934*a^3*c^2*x^2-844*a^3*c*x^3+664*a^3*x^4+59*a^2*b^5-41*a^2*b^4*c+
    
46. 320*a^2*b^4*x-98*a^2*b^3*c^2+2128*a^2*b^3*c*x+2270*a^2*b^3*x^2-98*a^2*b^2*c^3+1568*a^2*b^2*c^2*x+3594*a^2*b^2*c*x^2-2312*a^2*b^2*x^3-41*a^2*b*c^4+2128*a^2*b*c^3*x+3594*a^2*b*c^2*x^2-
    
47. 9952*a^2*b*c*x^3-12424*a^2*b*x^4+59*a^2*c^5+320*a^2*c^4*x+2270*a^2*c^3*x^2-2312*a^2*c^2*x^3-12424*a^2*c*x^4-4864*a^2*x^5-46*a*b^6-68*a*b^5*c+366*a*b^5*x+142*a*b^4*c^2-570*a*b^4*c*x-
    
48. 341*a*b^4*x^2+328*a*b^3*c^3-468*a*b^3*c^2*x-7028*a*b^3*c*x^2-5796*a*b^3*x^3+142*a*b^2*c^4-468*a*b^2*c^3*x+386*a*b^2*c^2*x^2+9364*a*b^2*c*x^3+6360*a*b^2*x^4-68*a*b*c^5-570*a*b*c^4*x-
    
49. 7028*a*b*c^3*x^2+9364*a*b*c^2*x^3+44448*a*b*c*x^4+18432*a*b*x^5-46*a*c^6+366*a*c^5*x-341*a*c^4*x^2-5796*a*c^3*x^3+6360*a*c^2*x^4+18432*a*c*x^5+3056*a*x^6-26*b^7-38*b^6*c-240*b^6*x-
    
50. 82*b^5*c^2+224*b^5*c*x+1125*b^5*x^2-238*b^4*c^3+752*b^4*c^2*x+5673*b^4*c*x^2+5094*b^4*x^3-238*b^3*c^4+576*b^3*c^3*x+5698*b^3*c^2*x^2+5096*b^3*c*x^3-744*b^3*x^4-82*b^2*c^5+
    
51. 752*b^2*c^4*x+5698*b^2*c^3*x^2-13820*b^2*c^2*x^3-46984*b^2*c*x^4-20992*b^2*x^5-38*b*c^6+224*b*c^5*x+5673*b*c^4*x^2+5096*b*c^3*x^3-46984*b*c^2*x^4-73824*b*c*x^5-18640*b*x^6-26*c^7-
    
52. 240*c^6*x+1125*c^5*x^2+5094*c^4*x^3-744*c^3*x^4-20992*c^2*x^5-18640*c*x^6-672*x^7)*L^5+(12*a^8+9*a^7*b+9*a^7*c-7*a^7*x+6*a^6*b^2+188*a^6*b*c+261*a^6*b*x+6*a^6*c^2+261*a^6*c*x-65*a^6*x^2-
    
53. 52*a^5*b^3-92*a^5*b^2*c-172*a^5*b^2*x-92*a^5*b*c^2+408*a^5*b*c*x+1137*a^5*b*x^2-52*a^5*c^3-172*a^5*c^2*x+1137*a^5*c*x^2+1177*a^5*x^3+25*a^4*b^4-396*a^4*b^3*c-184*a^4*b^3*x-74*a^4*b^2*c^2+
    
54. 776*a^4*b^2*c*x+432*a^4*b^2*x^2-396*a^4*b*c^3+776*a^4*b*c^2*x-96*a^4*b*c*x^2-743*a^4*b*x^3+25*a^4*c^4-184*a^4*c^3*x+432*a^4*c^2*x^2-743*a^4*c*x^3+144*a^4*x^4-67*a^3*b^5-111*a^3*b^4*c-
    
55. 375*a^3*b^4*x+66*a^3*b^3*c^2-2828*a^3*b^3*c*x-950*a^3*b^3*x^2+66*a^3*b^2*c^3-2666*a^3*b^2*c^2*x+94*a^3*b^2*c*x^2+3290*a^3*b^2*x^3-111*a^3*b*c^4-2828*a^3*b*c^3*x+94*a^3*b*c^2*x^2+
    
56. 7908*a^3*b*c*x^3+1444*a^3*b*x^4-67*a^3*c^5-375*a^3*c^4*x-950*a^3*c^3*x^2+3290*a^3*c^2*x^3+1444*a^3*c*x^4-232*a^3*x^5+36*a^2*b^6+552*a^2*b^5*c+189*a^2*b^5*x-36*a^2*b^4*c^2+65*a^2*b^4*c*x-
    
57. 1633*a^2*b^4*x^2-80*a^2*b^3*c^3+978*a^2*b^3*c^2*x-4548*a^2*b^3*c*x^2-5278*a^2*b^3*x^3-36*a^2*b^2*c^4+978*a^2*b^2*c^3*x-9414*a^2*b^2*c^2*x^2-12714*a^2*b^2*c*x^3+4212*a^2*b^2*x^4+
    
58. 552*a^2*b*c^5+65*a^2*b*c^4*x-4548*a^2*b*c^3*x^2-12714*a^2*b*c^2*x^3+4040*a^2*b*c*x^4+10288*a^2*b*x^5+36*a^2*c^6+189*a^2*c^5*x-1633*a^2*c^4*x^2-5278*a^2*c^3*x^3+4212*a^2*c^2*x^4+
    
59. 10288*a^2*c*x^5-584*a^2*x^6+46*a*b^7-110*a*b^6*c+186*a*b^6*x-170*a*b^5*c^2-500*a*b^5*c*x-1355*a*b^5*x^2+362*a*b^4*c^3+166*a*b^4*c^2*x+3593*a*b^4*c*x^2+2221*a*b^4*x^3+362*a*b^3*c^4-
    
60. 2392*a*b^3*c^3*x+466*a*b^3*c^2*x^2+15492*a*b^3*c*x^3+9164*a*b^3*x^4-170*a*b^2*c^5+166*a*b^2*c^4*x+466*a*b^2*c^3*x^2-3602*a*b^2*c^2*x^3-21132*a*b^2*c*x^4-15928*a*b^2*x^5-110*a*b*c^6-
    
61. 500*a*b*c^5*x+3593*a*b*c^4*x^2+15492*a*b*c^3*x^3-21132*a*b*c^2*x^4-52320*a*b*c*x^5-14816*a*b*x^6+46*a*c^7+186*a*c^6*x-1355*a*c^5*x^2+2221*a*c^4*x^3+9164*a*c^3*x^4-15928*a*c^2*x^5-
    
62. 14816*a*c*x^6+2352*a*x^7-15*b^8+8*b^7*c+198*b^7*x+92*b^6*c^2+506*b^6*c*x+642*b^6*x^2+56*b^5*c^3+782*b^5*c^2*x-1140*b^5*c*x^2-2779*b^5*x^3-26*b^4*c^4+1202*b^4*c^3*x-4770*b^4*c^2*x^2-
    
63. 17207*b^4*c*x^3-8244*b^4*x^4+56*b^3*c^5+1202*b^3*c^4*x-1880*b^3*c^3*x^2-13342*b^3*c^2*x^3-7424*b^3*c*x^4+3056*b^3*x^5+92*b^2*c^6+782*b^2*c^5*x-4770*b^2*c^4*x^2-13342*b^2*c^3*x^3+
    
64. 44456*b^2*c^2*x^4+86208*b^2*c*x^5+23368*b^2*x^6+8*b*c^7+506*b*c^6*x-1140*b*c^5*x^2-17207*b*c^4*x^3-7424*b*c^3*x^4+86208*b*c^2*x^5+85072*b*c*x^6+6960*b*x^7-15*c^8+198*c^7*x+
    
65. 642*c^6*x^2-2779*c^5*x^3-8244*c^4*x^4+3056*c^3*x^5+23368*c^2*x^6+6960*c*x^7-4320*x^8)*L^4+(-4*a^9-12*a^8*b-12*a^8*c-72*a^8*x+10*a^7*b^2-60*a^7*b*c-52*a^7*b*x+10*a^7*c^2-52*a^7*c*x-62*a^7*x^2+
    
66. 86*a^6*b^3+82*a^6*b^2*c+52*a^6*b^2*x+82*a^6*b*c^2-568*a^6*b*c*x-706*a^6*b*x^2+86*a^6*c^3+52*a^6*c^2*x-706*a^6*c*x^2-224*a^6*x^3-69*a^5*b^4+284*a^5*b^3*c+536*a^5*b^3*x+194*a^5*b^2*c^2+
    
67. 872*a^5*b^2*c*x+1016*a^5*b^2*x^2+284*a^5*b*c^3+872*a^5*b*c^2*x+560*a^5*b*c*x^2-756*a^5*b*x^3-69*a^5*c^4+536*a^5*c^3*x+1016*a^5*c^2*x^2-756*a^5*c*x^3-439*a^5*x^4-39*a^4*b^5+173*a^4*b^4*c+
    
68. 86*a^4*b^4*x-54*a^4*b^3*c^2+1272*a^4*b^3*c*x+984*a^4*b^3*x^2-54*a^4*b^2*c^3+1348*a^4*b^2*c^2*x+744*a^4*b^2*c*x^2-8*a^4*b^2*x^3+173*a^4*b*c^4+1272*a^4*b*c^3*x+744*a^4*b*c^2*x^2+
    
69. 3216*a^4*b*c*x^3+1827*a^4*b*x^4-39*a^4*c^5+86*a^4*c^4*x+984*a^4*c^3*x^2-8*a^4*c^2*x^3+1827*a^4*c*x^4+622*a^4*x^5+20*a^3*b^6-616*a^3*b^5*c-328*a^3*b^5*x-468*a^3*b^4*c^2-40*a^3*b^4*c*x+
    
70. 466*a^3*b^4*x^2-688*a^3*b^3*c^3+240*a^3*b^3*c^2*x+7592*a^3*b^3*c*x^2+2960*a^3*b^3*x^3-468*a^3*b^2*c^4+240*a^3*b^2*c^3*x+10604*a^3*b^2*c^2*x^2+6640*a^3*b^2*c*x^3-1754*a^3*b^2*x^4-
    
71. 616*a^3*b*c^5-40*a^3*b*c^4*x+7592*a^3*b*c^3*x^2+6640*a^3*b*c^2*x^3+28*a^3*b*c*x^4+2092*a^3*b*x^5+20*a^3*c^6-328*a^3*c^5*x+466*a^3*c^4*x^2+2960*a^3*c^3*x^3-1754*a^3*c^2*x^4+2092*a^3*c*x^5+
    
72. 200*a^3*x^6-44*a^2*b^7-68*a^2*b^6*c-180*a^2*b^6*x+404*a^2*b^5*c^2-1096*a^2*b^5*c*x-690*a^2*b^5*x^2+220*a^2*b^4*c^3-1740*a^2*b^4*c^2*x-3066*a^2*b^4*c*x^2+3128*a^2*b^4*x^3+220*a^2*b^3*c^4+
    
73. 400*a^2*b^3*c^3*x-2484*a^2*b^3*c^2*x^2+5120*a^2*b^3*c*x^3+6322*a^2*b^3*x^4+404*a^2*b^2*c^5-1740*a^2*b^2*c^4*x-2484*a^2*b^2*c^3*x^2+26384*a^2*b^2*c^2*x^3+27094*a^2*b^2*c*x^4-
    
74. 2216*a^2*b^2*x^5-68*a^2*b*c^6-1096*a^2*b*c^5*x-3066*a^2*b*c^4*x^2+5120*a^2*b*c^3*x^3+27094*a^2*b*c^2*x^4+4672*a^2*b*c*x^5-3112*a^2*b*x^6-44*a^2*c^7-180*a^2*c^6*x-690*a^2*c^5*x^2+3128*a^2*c^4*x^3+
    
75. 6322*a^2*c^3*x^4-2216*a^2*c^2*x^5-3112*a^2*c*x^6+5040*a^2*x^7+43*a*b^8+168*a*b^7*c-188*a*b^7*x+148*a*b^6*c^2+540*a*b^6*c*x-60*a*b^6*x^2-232*a*b^5*c^3-300*a*b^5*c^2*x+872*a*b^5*c*x^2+
    
76. 1908*a*b^5*x^3-510*a*b^4*c^4-820*a*b^4*c^3*x+764*a*b^4*c^2*x^2-7356*a*b^4*c*x^3-7487*a*b^4*x^4-232*a*b^3*c^5-820*a*b^3*c^4*x+5808*a*b^3*c^3*x^2+648*a*b^3*c^2*x^3-15388*a*b^3*c*x^4-
    
77. 6252*a*b^3*x^5+148*a*b^2*c^6-300*a*b^2*c^5*x+764*a*b^2*c^4*x^2+648*a*b^2*c^3*x^3+710*a*b^2*c^2*x^4+25564*a*b^2*c*x^5+26408*a*b^2*x^6+168*a*b*c^7+540*a*b*c^6*x+872*a*b*c^5*x^2-
    
78. 7356*a*b*c^4*x^3-15388*a*b*c^3*x^4+25564*a*b*c^2*x^5+27360*a*b*c*x^6+4560*a*b*x^7+43*a*c^8-188*a*c^7*x-60*a*c^6*x^2+1908*a*c^5*x^3-7487*a*c^4*x^4-6252*a*c^3*x^5+26408*a*c^2*x^6+
    
79. 4560*a*c*x^7-5616*a*x^8+9*b^9-15*b^8*c+18*b^8*x-92*b^7*c^2-288*b^7*c*x-564*b^7*x^2+20*b^6*c^3-808*b^6*c^2*x-1884*b^6*c*x^2-736*b^6*x^3+334*b^5*c^4-352*b^5*c^3*x-1460*b^5*c^2*x^2+
    
80. 4192*b^5*c*x^3+4411*b^5*x^4+334*b^4*c^5+300*b^4*c^4*x-3004*b^4*c^3*x^2+12768*b^4*c^2*x^3+30231*b^4*c*x^4+8410*b^4*x^5+20*b^3*c^6-352*b^3*c^5*x-3004*b^3*c^4*x^2+3392*b^3*c^3*x^3+
    
81. 14494*b^3*c^2*x^4-1576*b^3*c*x^5-7240*b^3*x^6-92*b^2*c^7-808*b^2*c^6*x-1460*b^2*c^5*x^2+12768*b^2*c^4*x^3+14494*b^2*c^3*x^4-76292*b^2*c^2*x^5-95176*b^2*c*x^6-17856*b^2*x^7-15*b*c^8-
    
82. 288*b*c^7*x-1884*b*c^6*x^2+4192*b*c^5*x^3+30231*b*c^4*x^4-1576*b*c^3*x^5-95176*b*c^2*x^6-40224*b*c*x^7+7344*b*x^8+9*c^9+18*c^8*x-564*c^7*x^2-736*c^6*x^3+4411*c^5*x^4+8410*c^4*x^5-
    
83. 7240*c^3*x^6-17856*c^2*x^7+7344*c*x^8+4320*x^9)*L^3+(4*a^9*b+4*a^9*c+20*a^9*x-12*a^8*b^2-24*a^8*b*c-12*a^8*b*x-12*a^8*c^2-12*a^8*c*x+96*a^8*x^2-38*a^7*b^3-34*a^7*b^2*c-106*a^7*b^2*x-
    
84. 34*a^7*b*c^2+60*a^7*b*c*x-34*a^7*b*x^2-38*a^7*c^3-106*a^7*c^2*x-34*a^7*c*x^2+34*a^7*x^3+23*a^6*b^4-84*a^6*b^3*c-290*a^6*b^3*x-214*a^6*b^2*c^2-918*a^6*b^2*c*x-824*a^6*b^2*x^2-84*a^6*b*c^3-
    
85. 918*a^6*b*c^2*x-304*a^6*b*c*x^2+18*a^6*b*x^3+23*a^6*c^4-290*a^6*c^3*x-824*a^6*c^2*x^2+18*a^6*c*x^3-239*a^6*x^4+75*a^5*b^5+7*a^5*b^4*c+269*a^5*b^4*x+158*a^5*b^3*c^2-572*a^5*b^3*c*x-
    
86. 1128*a^5*b^3*x^2+158*a^5*b^2*c^3-658*a^5*b^2*c^2*x-3352*a^5*b^2*c*x^2-2480*a^5*b^2*x^3+7*a^5*b*c^4-572*a^5*b*c^3*x-3352*a^5*b*c^2*x^2-4480*a^5*b*c*x^3-1155*a^5*b*x^4+75*a^5*c^5+
    
87. 269*a^5*c^4*x-1128*a^5*c^3*x^2-2480*a^5*c^2*x^3-1155*a^5*c*x^4-1245*a^5*x^5-38*a^4*b^6+252*a^4*b^5*c+457*a^4*b^5*x+582*a^4*b^4*c^2+701*a^4*b^4*c*x+544*a^4*b^4*x^2+584*a^4*b^3*c^3-
    
88. 438*a^4*b^3*c^2*x-3072*a^4*b^3*c*x^2-1784*a^4*b^3*x^3+582*a^4*b^2*c^4-438*a^4*b^2*c^3*x-4416*a^4*b^2*c^2*x^2-6792*a^4*b^2*c*x^3-2228*a^4*b^2*x^4+252*a^4*b*c^5+701*a^4*b*c^4*x-
    
89. 3072*a^4*b*c^3*x^2-6792*a^4*b*c^2*x^3-9688*a^4*b*c*x^4-2201*a^4*b*x^5-38*a^4*c^6+457*a^4*c^5*x+544*a^4*c^4*x^2-1784*a^4*c^3*x^3-2228*a^4*c^2*x^4-2201*a^4*c*x^5+50*a^4*x^6+
    
90. 4*a^3*b^7+124*a^3*b^6*c-120*a^3*b^6*x-332*a^3*b^5*c^2+1600*a^3*b^5*c*x+1354*a^3*b^5*x^2-564*a^3*b^4*c^3+2488*a^3*b^4*c^2*x+2002*a^3*b^4*c*x^2-326*a^3*b^4*x^3-564*a^3*b^3*c^4+
    
91. 2560*a^3*b^3*c^3*x-188*a^3*b^3*c^2*x^2-10136*a^3*b^3*c*x^3-4154*a^3*b^3*x^4-332*a^3*b^2*c^5+2488*a^3*b^2*c^4*x-188*a^3*b^2*c^3*x^2-21860*a^3*b^2*c^2*x^3-15678*a^3*b^2*c*x^4-
    
92. 818*a^3*b^2*x^5+124*a^3*b*c^6+1600*a^3*b*c^5*x+2002*a^3*b*c^4*x^2-10136*a^3*b*c^3*x^3-15678*a^3*b*c^2*x^4-11140*a^3*b*c*x^5-4556*a^3*b*x^6+4*a^3*c^7-120*a^3*c^6*x+1354*a^3*c^5*x^2-
    
93. 326*a^3*c^4*x^3-4154*a^3*c^3*x^4-818*a^3*c^2*x^5-4556*a^3*c*x^6+1464*a^3*x^7+3*a^2*b^8-264*a^2*b^7*c-144*a^2*b^7*x-300*a^2*b^6*c^2-352*a^2*b^6*c*x+404*a^2*b^6*x^2+200*a^2*b^5*c^3-
    
94. 384*a^2*b^5*c^2*x+2616*a^2*b^5*c*x^2+1610*a^2*b^5*x^3+466*a^2*b^4*c^4-1424*a^2*b^4*c^3*x+4332*a^2*b^4*c^2*x^2+8690*a^2*b^4*c*x^3-2531*a^2*b^4*x^4+200*a^2*b^3*c^5-1424*a^2*b^3*c^4*x+
    
95. 144*a^2*b^3*c^3*x^2+2596*a^2*b^3*c^2*x^3-8796*a^2*b^3*c*x^4-5730*a^2*b^3*x^5-300*a^2*b^2*c^6-384*a^2*b^2*c^5*x+4332*a^2*b^2*c^4*x^2+2596*a^2*b^2*c^3*x^3-35890*a^2*b^2*c^2*x^4-
    
96. 30854*a^2*b^2*c*x^5-2552*a^2*b^2*x^6-264*a^2*b*c^7-352*a^2*b*c^6*x+2616*a^2*b*c^5*x^2+8690*a^2*b*c^4*x^3-8796*a^2*b*c^3*x^4-30854*a^2*b*c^2*x^5+4912*a^2*b*c*x^6-552*a^2*b*x^7+
    
97. 3*a^2*c^8-144*a^2*c^7*x+404*a^2*c^6*x^2+1610*a^2*c^5*x^3-2531*a^2*c^4*x^4-5730*a^2*c^3*x^5-2552*a^2*c^2*x^6-552*a^2*c*x^7-5004*a^2*x^8-21*a*b^9+19*a*b^8*c-47*a*b^8*x+188*a*b^7*c^2-
    
98. 56*a*b^7*c*x+400*a*b^7*x^2+44*a*b^6*c^3+220*a*b^6*c^2*x-544*a*b^6*c*x^2-716*a*b^6*x^3-486*a*b^5*c^4+632*a*b^5*c^3*x+576*a*b^5*c^2*x^2-2136*a*b^5*c*x^3-1267*a*b^5*x^4-486*a*b^4*c^5+
    
99. 806*a*b^4*c^4*x+1104*a*b^4*c^3*x^2-3444*a*b^4*c^2*x^3+5185*a*b^4*c*x^4+10559*a*b^4*x^5+44*a*b^3*c^6+632*a*b^3*c^5*x+1104*a*b^3*c^4*x^2-6096*a*b^3*c^3*x^3+1794*a*b^3*c^2*x^4+
    
100. 10028*a*b^3*c*x^5+28*a*b^3*x^6+188*a*b^2*c^7+220*a*b^2*c^6*x+576*a*b^2*c^5*x^2-3444*a*b^2*c^4*x^3+1794*a*b^2*c^3*x^4+4570*a*b^2*c^2*x^5-19100*a*b^2*c*x^6-22584*a*b^2*x^7+19*a*b*c^8-
     
101. 56*a*b*c^7*x-544*a*b*c^6*x^2-2136*a*b*c^5*x^3+5185*a*b*c^4*x^4+10028*a*b*c^3*x^5-19100*a*b*c^2*x^6-5184*a*b*c*x^7+2808*a*b*x^8-21*a*c^9-47*a*c^8*x+400*a*c^7*x^2-716*a*c^6*x^3-1267*a*c^5*x^4+
     
102. 10559*a*c^4*x^5+28*a*c^3*x^6-22584*a*c^2*x^7+2808*a*c*x^8+4320*a*x^9-27*b^9*x+45*b^8*c*x+36*b^8*x^2+276*b^7*c^2*x+816*b^7*c*x^2+764*b^7*x^3-60*b^6*c^3*x+1872*b^6*c^2*x^2+2996*b^6*c*x^3+
     
103. 246*b^6*x^4-1002*b^5*c^4*x+720*b^5*c^3*x^2+508*b^5*c^2*x^3-7916*b^5*c*x^4-4629*b^5*x^5-1002*b^4*c^5*x-744*b^4*c^4*x^2+4180*b^4*c^3*x^3-16214*b^4*c^2*x^4-29913*b^4*c*x^5-5770*b^4*x^6-
     
104. 60*b^3*c^6*x+720*b^3*c^5*x^2+4180*b^3*c^4*x^3-3816*b^3*c^3*x^4-5026*b^3*c^2*x^5+16168*b^3*c*x^6+9384*b^3*x^7+276*b^2*c^7*x+1872*b^2*c^6*x^2+508*b^2*c^5*x^3-16214*b^2*c^4*x^4-
     
105. 5026*b^2*c^3*x^5+70180*b^2*c^2*x^6+58056*b^2*c*x^7+11412*b^2*x^8+45*b*c^8*x+816*b*c^7*x^2+2996*b*c^6*x^3-7916*b*c^5*x^4-29913*b*c^4*x^5+16168*b*c^3*x^6+58056*b*c^2*x^7-16632*b*c*x^8-
     
106. 8640*b*x^9-27*c^9*x+36*c^8*x^2+764*c^7*x^3+246*c^6*x^4-4629*c^5*x^5-5770*c^4*x^6+9384*c^3*x^7+11412*c^2*x^8-8640*c*x^9-1296*x^10)*L^2+(4*a^9*b^2+8*a^9*b*c+8*a^9*b*x+4*a^9*c^2+8*a^9*c*x-
     
107. 12*a^9*x^2+12*a^8*b^3+36*a^8*b^2*c+96*a^8*b^2*x+36*a^8*b*c^2+192*a^8*b*c*x+204*a^8*b*x^2+12*a^8*c^3+96*a^8*c^2*x+204*a^8*c*x^2+72*a^8*x^3-15*a^7*b^4+20*a^7*b^3*c+76*a^7*b^3*x+
     
108. 70*a^7*b^2*c^2+260*a^7*b^2*c*x+398*a^7*b^2*x^2+20*a^7*b*c^3+260*a^7*b*c^2*x+236*a^7*b*c*x^2+380*a^7*b*x^3-15*a^7*c^4+76*a^7*c^3*x+398*a^7*c^2*x^2+380*a^7*c*x^3+201*a^7*x^4-69*a^6*b^5-
     
109. 169*a^6*b^4*c-232*a^6*b^4*x-162*a^6*b^3*c^2-32*a^6*b^3*c*x+298*a^6*b^3*x^2-162*a^6*b^2*c^3+400*a^6*b^2*c^2*x+2414*a^6*b^2*c*x^2+1612*a^6*b^2*x^3-169*a^6*b*c^4-32*a^6*b*c^3*x+
     
110. 2414*a^6*b*c^2*x^2+2872*a^6*b*c*x^3+723*a^6*b*x^4-69*a^6*c^5-232*a^6*c^4*x+298*a^6*c^3*x^2+1612*a^6*c^2*x^3+723*a^6*c*x^4+84*a^6*x^5+58*a^5*b^6-52*a^5*b^5*c-338*a^5*b^5*x-
     
111. 218*a^5*b^4*c^2-410*a^5*b^4*c*x-707*a^5*b^4*x^2-216*a^5*b^3*c^3-52*a^5*b^3*c^2*x+900*a^5*b^3*c*x^2+1176*a^5*b^3*x^3-218*a^5*b^2*c^4-52*a^5*b^2*c^3*x+910*a^5*b^2*c^2*x^2+
     
112. 4776*a^5*b^2*c*x^3+2680*a^5*b^2*x^4-52*a^5*b*c^5-410*a^5*b*c^4*x+900*a^5*b*c^3*x^2+4776*a^5*b*c^2*x^3+6608*a^5*b*c*x^4+1298*a^5*b*x^5+58*a^5*c^6-338*a^5*c^5*x-707*a^5*c^4*x^2+
     
113. 1176*a^5*c^3*x^3+2680*a^5*c^2*x^4+1298*a^5*c*x^5+585*a^5*x^6+70*a^4*b^7+74*a^4*b^6*c+68*a^4*b^6*x+158*a^4*b^5*c^2-760*a^4*b^5*c*x-949*a^4*b^5*x^2+594*a^4*b^4*c^3-1604*a^4*b^4*c^2*x-
     
114. 3065*a^4*b^4*c*x^2-790*a^4*b^4*x^3+594*a^4*b^3*c^4-1552*a^4*b^3*c^3*x-354*a^4*b^3*c^2*x^2+4360*a^4*b^3*c*x^3+2472*a^4*b^3*x^4+158*a^4*b^2*c^5-1604*a^4*b^2*c^4*x-354*a^4*b^2*c^3*x^2+
     
115. 8252*a^4*b^2*c^2*x^3+10856*a^4*b^2*c*x^4+1620*a^4*b^2*x^5+74*a^4*b*c^6-760*a^4*b*c^5*x-3065*a^4*b*c^4*x^2+4360*a^4*b*c^3*x^3+10856*a^4*b*c^2*x^4+8120*a^4*b*c*x^5-273*a^4*b*x^6+
     
116. 70*a^4*c^7+68*a^4*c^6*x-949*a^4*c^5*x^2-790*a^4*c^4*x^3+2472*a^4*c^3*x^4+1620*a^4*c^2*x^5-273*a^4*c*x^6-570*a^4*x^7-55*a^3*b^8+88*a^3*b^7*c+300*a^3*b^7*x+156*a^3*b^6*c^2-
     
117. 76*a^3*b^6*c*x-12*a^3*b^6*x^2-24*a^3*b^5*c^3+284*a^3*b^5*c^2*x-2104*a^3*b^5*c*x^2-2176*a^3*b^5*x^3-74*a^3*b^4*c^4+1284*a^3*b^4*c^3*x-3380*a^3*b^4*c^2*x^2-3776*a^3*b^4*c*x^3+
     
118. 1189*a^3*b^4*x^4-24*a^3*b^3*c^5+1284*a^3*b^3*c^4*x-3600*a^3*b^3*c^3*x^2+192*a^3*b^3*c^2*x^3+8116*a^3*b^3*c*x^4+3248*a^3*b^3*x^5+156*a^3*b^2*c^6+284*a^3*b^2*c^5*x-3380*a^3*b^2*c^4*x^2+
     
119. 192*a^3*b^2*c^3*x^3+20510*a^3*b^2*c^2*x^4+14256*a^3*b^2*c*x^5+534*a^3*b^2*x^6+88*a^3*b*c^7-76*a^3*b*c^6*x-2104*a^3*b*c^5*x^2-3776*a^3*b*c^4*x^3+8116*a^3*b*c^3*x^4+14256*a^3*b*c^2*x^5+
     
120. 7116*a^3*b*c*x^6+2172*a^3*b*x^7-55*a^3*c^8+300*a^3*c^7*x-12*a^3*c^6*x^2-2176*a^3*c^5*x^3+1189*a^3*c^4*x^4+3248*a^3*c^3*x^5+534*a^3*c^2*x^6+2172*a^3*c*x^7-1440*a^3*x^8-5*a^2*b^9+
     
121. 19*a^2*b^8*c+28*a^2*b^8*x-180*a^2*b^7*c^2+272*a^2*b^7*c*x+164*a^2*b^7*x^2-292*a^2*b^6*c^3+560*a^2*b^6*c^2*x+860*a^2*b^6*c*x^2-460*a^2*b^6*x^3+202*a^2*b^5*c^4+240*a^2*b^5*c^3*x+
     
122. 20*a^2*b^5*c^2*x^2-2872*a^2*b^5*c*x^3-825*a^2*b^5*x^4+202*a^2*b^4*c^5-152*a^2*b^4*c^4*x+2028*a^2*b^4*c^3*x^2-5300*a^2*b^4*c^2*x^3-8077*a^2*b^4*c*x^4+1032*a^2*b^4*x^5-292*a^2*b^3*c^6+
     
123. 240*a^2*b^3*c^5*x+2028*a^2*b^3*c^4*x^2-3728*a^2*b^3*c^3*x^3-1162*a^2*b^3*c^2*x^4+10672*a^2*b^3*c*x^5+2154*a^2*b^3*x^6-180*a^2*b^2*c^7+560*a^2*b^2*c^6*x+20*a^2*b^2*c^5*x^2-
     
124. 5300*a^2*b^2*c^4*x^3-1162*a^2*b^2*c^3*x^4+27600*a^2*b^2*c^2*x^5+16878*a^2*b^2*c*x^6+2568*a^2*b^2*x^7+19*a^2*b*c^8+272*a^2*b*c^7*x+860*a^2*b*c^6*x^2-2872*a^2*b*c^5*x^3-8077*a^2*b*c^4*x^4+
     
125. 10672*a^2*b*c^3*x^5+16878*a^2*b*c^2*x^6-13920*a^2*b*c*x^7+1368*a^2*b*x^8-5*a^2*c^9+28*a^2*c^8*x+164*a^2*c^7*x^2-460*a^2*c^6*x^3-825*a^2*c^5*x^4+1032*a^2*c^4*x^5+2154*a^2*c^3*x^6+
     
126. 2568*a^2*c^2*x^7+1368*a^2*c*x^8+2376*a^2*x^9+42*a*b^9*x-38*a*b^8*c*x-35*a*b^8*x^2-376*a*b^7*c^2*x-392*a*b^7*c*x^2-420*a*b^7*x^3-88*a*b^6*c^3*x-884*a*b^6*c^2*x^2-92*a*b^6*c*x^3+
     
127. 1098*a*b^6*x^4+972*a*b^5*c^4*x-568*a*b^5*c^3*x^2+428*a*b^5*c^2*x^3+3996*a*b^5*c*x^4+766*a*b^5*x^5+972*a*b^4*c^5*x-82*a*b^4*c^4*x^2-1196*a*b^4*c^3*x^3+3222*a*b^4*c^2*x^4-298*a*b^4*c*x^5-
     
128. 6495*a*b^4*x^6-88*a*b^3*c^6*x-568*a*b^3*c^5*x^2-1196*a*b^3*c^4*x^3+2696*a*b^3*c^3*x^4-5236*a*b^3*c^2*x^5-8220*a*b^3*c*x^6+1236*a*b^3*x^7-376*a*b^2*c^7*x-884*a*b^2*c^6*x^2+428*a*b^2*c^5*x^3+
     
129. 3222*a*b^2*c^4*x^4-5236*a*b^2*c^3*x^5-954*a*b^2*c^2*x^6+10812*a*b^2*c*x^7+7416*a*b^2*x^8-38*a*b*c^8*x-392*a*b*c^7*x^2-92*a*b*c^6*x^3+3996*a*b*c^5*x^4-298*a*b*c^4*x^5-8220*a*b*c^3*x^6+
     
130. 10812*a*b*c^2*x^7+5184*a*b*c*x^8-3672*a*b*x^9+42*a*c^9*x-35*a*c^8*x^2-420*a*c^7*x^3+1098*a*c^6*x^4+766*a*c^5*x^5-6495*a*c^4*x^6+1236*a*c^3*x^7+7416*a*c^2*x^8-3672*a*c*x^9-1296*a*x^10+
     
131. 27*b^9*x^2-45*b^8*c*x^2-66*b^8*x^3-276*b^7*c^2*x^2-800*b^7*c*x^3-498*b^7*x^4+60*b^6*c^3*x^2-1688*b^6*c^2*x^3-2174*b^6*c*x^4+144*b^6*x^5+1002*b^5*c^4*x^2-608*b^5*c^3*x^3+646*b^5*c^2*x^4+
     
132. 6720*b^5*c*x^5+2823*b^5*x^6+1002*b^4*c^5*x^2+692*b^4*c^4*x^3-2966*b^4*c^3*x^4+9840*b^4*c^2*x^5+15411*b^4*c*x^6+2562*b^4*x^7+60*b^3*c^6*x^2-608*b^3*c^5*x^3-2966*b^3*c^4*x^4+
     
133. 2432*b^3*c^3*x^5-2250*b^3*c^2*x^6-16008*b^3*c*x^7-6192*b^3*x^8-276*b^2*c^7*x^2-1688*b^2*c^6*x^3+646*b^2*c^5*x^4+9840*b^2*c^4*x^5-2250*b^2*c^3*x^6-32532*b^2*c^2*x^7-16128*b^2*c*x^8-
     
134. 6048*b^2*x^9-45*b*c^8*x^2-800*b*c^7*x^3-2174*b*c^6*x^4+6720*b*c^5*x^5+15411*b*c^4*x^6-16008*b*c^3*x^7-16128*b*c^2*x^8+26784*b*c*x^9+2592*b*x^10+27*c^9*x^2-66*c^8*x^3-498*c^7*x^4+
     
135. 144*c^6*x^5+2823*c^5*x^6+2562*c^4*x^7-6192*c^3*x^8-6048*c^2*x^9+2592*c*x^10)*L=0

复制代码

理论上来说：\(x\)关于\({a,b,c}\)的代数方程应该不会太复杂，至少应比\(r\)关于\({a,b,c}\)的代数方程要简单

例如：当\(b=c\)时，我得到 \(36x^4+(-24a-48b)x^3+(41a^2-8ab-84b^2)x^2+(4a^3-12a^2b-16ab^2+48b^3)x+a^4+4a^3b-8a^2b^2-32ab^3+48b^4=0\)

数学星空

注：天下无毒史先生曾给了一个网页链接，好像没有给出r,x,y,z关于{a,b,c}的任何有用的信息？

http://faculty.evansville.edu/ck6/integer/unsolved.html

19. Congruent Incircles Point
Reward: $50.00
Noam Elkies proved that there is a point X in the plane of an arbitrary ABC such that the triangles AXB, BXC, CXA have congruent incircles. Find reasonable barycentric coordinates for X. (The point X is listed as X(5394) in the Encyclopedia of Triangle Centers: ETC - Part 3)




下面是 http://faculty.evansville.edu/ck6/encyclopedia/ETCPart4.html  网页的内容：

X(5394) = CONGRUENT INCIRCLES POINT
Barycentrics   (unknown)
X(5394) is the point X for which the three triangles AXB, BXC, CXA have congruent incircles. The existence of this point is proved by Noam Elkies in Mathematics Magazine 60 (1987) 117. His proof applies to a much wider range of functions (with the inradius replaced by the area, semiperimeter, etc., or any positive combination thereof).

Following is a copy-and-run Mathematica program that computes actual trilinear distances (1.7916..., 1.7057..., 1.6328...) of X(5394) for the triangle given by (a,b,c) = (6,9,13).



(1/2 Sqrt[(a + b - c) (a - b + c) (-a + b + c) (a + b + c)] {x/a, y/b, z/c} /. #1 /.
NSolve[{x + y + z == 1, (a + Sqrt[b^2 x^2 + (a^2 + b^2 - c^2) x y + a^2 y^2] +
Sqrt[c^2 x^2 + (a^2 - b^2 + c^2) x z + a^2 z^2])/
x == (b + Sqrt[b^2 x^2 + (a^2 + b^2 - c^2) x y + a^2 y^2] +
Sqrt[c^2 y^2 + (-a^2 + b^2 + c^2) y z + b^2 z^2])/
y == (c + Sqrt[c^2 x^2 + (a^2 - b^2 + c^2) x z + a^2 z^2] +
Sqrt[c^2 y^2 + (-a^2 + b^2 + c^2) y z + b^2 z^2])/
z /. #1}, {x, y, z}, WorkingPrecision -> 40][[1]] &)[Thread[{a, b, c} -> {6, 9,13}]] (* Program by Peter Moses, October 23, 2012. *)

X(5394) lies on no line X(i)X(j) for 1 <= i < j <= 5393.

数学星空

下面是  http://mathworld.wolfram.com/CongruentIncirclesPoint.html   给出的有关说明：

Congruent Incircles Point
   
The triangulation point \( Y\) of a reference triangle \(\triangle ABC\) for which triangles \(\triangle BYC, \triangle CYA\), and \(\triangle AYB\) have congruent incircles. It is a special case of an Elkies point. Kimberling and Elkies (1987) showed that a unique such point exists for any triangle, but did not provide explicit constructions for this point.

数学星空

我想到一个求解方案：上面提到当\(b=c\),得到了下面\(x\)关于\({a,b,c}\)的代数方程

\(36x^4+(-24a-48b)x^3+(41a^2-8ab-84b^2)x^2+(4a^3-12a^2b-16ab^2+48b^3)x+a^4+4a^3b-8a^2b^2-32ab^3+48b^4\)

\(=48b^4+16(3x-2a)b^3-8(a^2+2ax+21x^2)b^2+4(a^3-3a^2x-2ax^2-12x^3)b+a^4+4a^3x+41a^2x^2-24ax^3+36x^4\)

由于构成 \(b\)只可能是\(b+c\),因此可作代换\(b\to (b+c)/2\)

由于构成\(b^2\)只可能是\({b^2+c^2,bc}\),因此可作代换\(8b^2\to m_1(b^2+c^2)+(8-2m_1)bc\)

由于构成 \(b^3\)只可能是\({b^3+c^3,b^2c+bc^2}\),因此可作代换\(16b^3\to m_2(b^3+c^3)+(8-m_2)(b^2c+bc^2)\)

由于构成\( b^4\)只可能是\({b^4+c^4,b^3c+bc^3,b^2c^2}\),因此可作代换\(48b^4\to m_3(b^4+c^4)+m_4(b^3c+bc^3)+(48-2m_3-2m_4)b^2c^2\)

代入上面方程后可以得到

\(((-21b^2+42bc-21c^2)x^2+(-2ab^2+4abc-2ac^2)x-a^2b^2+2a^2bc-a^2c^2)m_1+((-6b^3+3b^2c+3bc^2-6c^3)x+4ab^3-2ab^2c-2abc^2+4ac^3)m_2+(-2a^2b^2+b^4+c^4)m_3+(-2a^2b^2+b^3c+bc^3)m_4+36x^4+(-24a-24b-24c)x^3+(41a^2-4ab-4ac-168bc)x^2+(4a^3-6a^2b-6a^2c-16abc+24b^3+24c^3)x+a^4+2a^3b+2a^3c+48a^2b^2-8a^2bc-16ab^3-16ac^3=0\)

一般来说：\(m_1,m_2,m_3,m_4\)只能取整数

我们通过数值计算可以得到：

\({a = 5, b = 12, c = 8, r =0 .5579288175, x = 7.508945269, y = 1.309969879, z = 4.712519498}\),
\({a = 5, b = 12, c = 9, r = 0.7457030106, x = 8.023198840, y = 1.936333889, z = 4.391943349}\),
\({a = 5, b = 12, c = 10, r =0.8598979637, x = 8.524477990, y = 2.457575118, z = 4.061568686}\),
\({a = 5, b = 12, c = 11, r =0.9270860023, x = 9.015533833, y = 2.924484659, z = 3.717914105}\),
\({a = 5, b = 12, c = 12, r =0.9554641421, x = 9.498901032, y = 3.355252602, z = 3.355252602}\),
\({a = 5, b = 12, c = 13, r = 0.9463673765, x = 9.976935430, y = 3.757971209, z = 2.965256262}\),
\({a = 5, b = 12, c = 15, r = 0.7930888457, x = 10.92873470, y = 4.488265076, z = 2.038266262}\)

那么我们现在的问题就变成找出满足上面解的整数\(m_1,m_2,m_3,m_4\),使数值解成立？

不知大家还有什么更好的方法得到\(x\)关于\({a,b,c}\)的代数方程 ？