Solving a system of non-linear equations with four unknowns and two parameters

Question

I need to solve a relatively simple system of 4 non-linear equations with 4 variables (a, b, p, q) and 2 parameters (λ, μ).

If I give numerical values to the 2 parameters, I easily get a numerical solution. However, it would be much better for my model if I could find an analytical solution where each of the four variables is expressed as a function of the two parameters.

I tried looking at answers to similar questions, but often the systems considered there were much more complicated and involved mathematical concepts I have not mastered. So I decided to post another question. If anyone would be so kind to help me, it would be very much appreciated!

Here is the system:

exp1 = a == 
1/3 (2 + 2 q - λ - Sqrt[
   4 + 3 b^2 - 12 p + 9 p^2 + 8 q - 6 b q - 12 p q + 7 q^2 + 
    2 λ + 2 q λ + λ^2]);
exp2 = p == 
1/6 (4 + 3 a + 2 q + λ - Sqrt[
   4 + 9 a^2 + 4 q - 12 a q + 4 q^2 - 4 λ + 
    6 a λ - 8 q λ + λ^2]);
exp3 = b == 
1/6 (2 + 2 a + 3 q - 2 μ - Sqrt[
   4 + 8 a + 4 a^2 - 12 q - 12 a q + 9 q^2 + 4 μ + 
    4 a μ + 4 μ^2]);
exp4 = q == 
1/3 (4 + 2 a - Sqrt[
   10 + 16 a + 7 a^2 - 12 b - 12 a b + 9 b^2 - 6 p - 6 a p + 
    3 p^2 - 6 μ - 6 a μ + 6 b μ]);

I know that all variables and parameters are positive, that p>a and q>b, and that if λ == μ, then p = a + λ = q = b + μ. Perhaps these facts can be used to get a solution.

I already tried:

Solve[{exp1, exp2, exp3, exp4}, {a, p, b, q}, Reals]

and (following an answer to a similar post)

v = Total[{exp1, exp2, exp3, exp4} /. Equal[l_, r_] -> Norm[l - (r)]];
NMinimize[v, {a, p, b, q}, MaxIterations -> 10^3]

but neither worked.

(I also tried to solve it myself by rearranging terms to get rid of the square roots and replacing expressions here and there, but could not get anywhere.)

Answer 1

One can obtain a Groebner basis for {a,b,p,q} in terms of the parameters. I do this in three steps below.

(1) Rewrite the radicals as new variables with defining equations of the form sqrt^2-radicand

(2) Eliminate these new variables using a suitable term order

(3) Form a lexicographic GB from the elimination ideal

polys = {-a + 1/3 (2 + 2 q - \[Lambda] - sqr1),
   -p + 1/6 (4 + 3 a + 2 q + \[Lambda] - sqr2),
   -b + 1/6 (2 + 2 a + 3 q - 2 \[Mu] - sqr3),
   -q + 1/3 (4 + 2 a - sqr4),
   sqr1^2 - (4 + 3 b^2 - 12 p + 9 p^2 + 8 q - 6 b q - 12 p q + 
      7 q^2 + 2 \[Lambda] + 2 q \[Lambda] + \[Lambda]^2),
   sqr2^2 - (4 + 9 a^2 + 4 q - 12 a q + 4 q^2 - 4 \[Lambda] + 
      6 a \[Lambda] - 8 q \[Lambda] + \[Lambda]^2), 
   sqr3^2 - (4 + 8 a + 4 a^2 - 12 q - 12 a q + 9 q^2 + 4 \[Mu] + 
      4 a \[Mu] + 4 \[Mu]^2), 
   sqr4^2 - (4 + 8 a + 4 a^2 - 12 q - 12 a q + 9 q^2 + 4 \[Mu] + 
      4 a \[Mu] + 4 \[Mu]^2)};

Timing[
 gb0 = GroebnerBasis[polys, {a, b, p, q}, {sqr1, sqr2, sqr3, sqr4}, 
    MonomialOrder -> EliminationOrder, 
    CoefficientDomain -> RationalFunctions];]

(* Out[7]= {0.0625, Null} *)

Timing[
 gb1 = GroebnerBasis[gb0, {a, b, p, q}, 
    CoefficientDomain -> RationalFunctions];]

(* Out[10]= {1.95313, Null} *)

The point is that now we have a univariate in the variable q (in terms of the parameters of course) so it can be solved in terms of those parameters. I won't show the full basis but the univariate is as below. All remaining polynomials are linear in their respective lead variables and thus solving for them in terms of solutions of q is trivial.

In[18]:= gb1[[1]]

(* Out[18]= 268130112 - 300231584 \[Lambda] + 96183560 \[Lambda]^2 + 
 2829952 \[Lambda]^3 - 5252040 \[Lambda]^4 + 217920 \[Lambda]^5 + 
 86400 \[Lambda]^6 - 746673528 \[Mu] + 874513072 \[Lambda] \[Mu] - 
 295906092 \[Lambda]^2 \[Mu] - 5831136 \[Lambda]^3 \[Mu] + 
 16556076 \[Lambda]^4 \[Mu] - 780288 \[Lambda]^5 \[Mu] - 
 285120 \[Lambda]^6 \[Mu] + 464772150 \[Mu]^2 - 
 700428416 \[Lambda] \[Mu]^2 + 299942288 \[Lambda]^2 \[Mu]^2 - 
 5186208 \[Lambda]^3 \[Mu]^2 - 18399270 \[Lambda]^4 \[Mu]^2 + 
 1233504 \[Lambda]^5 \[Mu]^2 + 368064 \[Lambda]^6 \[Mu]^2 + 
 459361293 \[Mu]^3 - 254180864 \[Lambda] \[Mu]^3 - 
 37881518 \[Lambda]^2 \[Mu]^3 + 22620472 \[Lambda]^3 \[Mu]^3 + 
 5622321 \[Lambda]^4 \[Mu]^3 - 1099872 \[Lambda]^5 \[Mu]^3 - 
 215136 \[Lambda]^6 \[Mu]^3 - 577143612 \[Mu]^4 + 
 585074396 \[Lambda] \[Mu]^4 - 131637868 \[Lambda]^2 \[Mu]^4 - 
 19484184 \[Lambda]^3 \[Mu]^4 + 4719324 \[Lambda]^4 \[Mu]^4 + 
 547956 \[Lambda]^5 \[Mu]^4 + 30456 \[Lambda]^6 \[Mu]^4 - 
 34075596 \[Mu]^5 - 121167612 \[Lambda] \[Mu]^5 + 
 75560994 \[Lambda]^2 \[Mu]^5 + 1720620 \[Lambda]^3 \[Mu]^5 - 
 4147218 \[Lambda]^4 \[Mu]^5 - 90648 \[Lambda]^5 \[Mu]^5 + 
 27540 \[Lambda]^6 \[Mu]^5 + 231557952 \[Mu]^6 - 
 152851560 \[Lambda] \[Mu]^6 + 6271428 \[Lambda]^2 \[Mu]^6 + 
 5712048 \[Lambda]^3 \[Mu]^6 + 635874 \[Lambda]^4 \[Mu]^6 - 
 53496 \[Lambda]^5 \[Mu]^6 - 12960 \[Lambda]^6 \[Mu]^6 - 
 33351060 \[Mu]^7 + 62452272 \[Lambda] \[Mu]^7 - 
 15696156 \[Lambda]^2 \[Mu]^7 - 2427036 \[Lambda]^3 \[Mu]^7 + 
 385341 \[Lambda]^4 \[Mu]^7 + 30888 \[Lambda]^5 \[Mu]^7 + 
 648 \[Lambda]^6 \[Mu]^7 - 45277056 \[Mu]^8 + 
 15983388 \[Lambda] \[Mu]^8 + 2779032 \[Lambda]^2 \[Mu]^8 - 
 217404 \[Lambda]^3 \[Mu]^8 - 141192 \[Lambda]^4 \[Mu]^8 - 
 3204 \[Lambda]^5 \[Mu]^8 + 648 \[Lambda]^6 \[Mu]^8 + 
 9482328 \[Mu]^9 - 9723656 \[Lambda] \[Mu]^9 + 
 947828 \[Lambda]^2 \[Mu]^9 + 276904 \[Lambda]^3 \[Mu]^9 - 
 1140 \[Lambda]^4 \[Mu]^9 - 1248 \[Lambda]^5 \[Mu]^9 - 
 108 \[Lambda]^6 \[Mu]^9 + 4691448 \[Mu]^10 - 
 532256 \[Lambda] \[Mu]^10 - 318672 \[Lambda]^2 \[Mu]^10 - 
 21324 \[Lambda]^3 \[Mu]^10 + 5592 \[Lambda]^4 \[Mu]^10 + 
 264 \[Lambda]^5 \[Mu]^10 - 951372 \[Mu]^11 + 
 604024 \[Lambda] \[Mu]^11 - 3364 \[Lambda]^2 \[Mu]^11 - 
 8544 \[Lambda]^3 \[Mu]^11 - 516 \[Lambda]^4 \[Mu]^11 - 
 245472 \[Mu]^12 - 4976 \[Lambda] \[Mu]^12 + 
 8920 \[Lambda]^2 \[Mu]^12 + 880 \[Lambda]^3 \[Mu]^12 + 
 34224 \[Mu]^13 - 11968 \[Lambda] \[Mu]^13 - 
 628 \[Lambda]^2 \[Mu]^13 + 5064 \[Mu]^14 + 264 \[Lambda] \[Mu]^14 - 
 156 \[Mu]^15 + 
 q^7 (135000 \[Mu]^2 + 249750 \[Mu]^3 + 167400 \[Mu]^4 + 
    49950 \[Mu]^5 + 6264 \[Mu]^6 + 216 \[Mu]^7) + 
 q^6 (2160000 + 3456000 \[Mu] - 487650 \[Mu]^2 + 
    178200 \[Lambda] \[Mu]^2 + 9450 \[Lambda]^2 \[Mu]^2 - 
    2363565 \[Mu]^3 + 247860 \[Lambda] \[Mu]^3 + 
    13635 \[Lambda]^2 \[Mu]^3 - 123768 \[Mu]^4 + 
    42444 \[Lambda] \[Mu]^4 + 1188 \[Lambda]^2 \[Mu]^4 + 
    910893 \[Mu]^5 - 55566 \[Lambda] \[Mu]^5 - 
    3672 \[Lambda]^2 \[Mu]^5 + 415074 \[Mu]^6 - 
    21384 \[Lambda] \[Mu]^6 - 702 \[Lambda]^2 \[Mu]^6 + 
    63450 \[Mu]^7 - 1782 \[Lambda] \[Mu]^7 - 27 \[Lambda]^2 \[Mu]^7 + 
    2508 \[Mu]^8 - 12 \[Mu]^9) + 
 q^5 (-29661600 + 3628800 \[Lambda] + 302400 \[Lambda]^2 - 
    25291080 \[Mu] + 3810240 \[Lambda] \[Mu] + 
    360720 \[Lambda]^2 \[Mu] + 25463460 \[Mu]^2 - 
    3312864 \[Lambda] \[Mu]^2 - 166428 \[Lambda]^2 \[Mu]^2 + 
    9936 \[Lambda]^3 \[Mu]^2 + 21212034 \[Mu]^3 - 
    2896632 \[Lambda] \[Mu]^3 - 160434 \[Lambda]^2 \[Mu]^3 + 
    9504 \[Lambda]^3 \[Mu]^3 - 8076894 \[Mu]^4 + 
    1624860 \[Lambda] \[Mu]^4 + 65610 \[Lambda]^2 \[Mu]^4 - 
    6588 \[Lambda]^3 \[Mu]^4 - 6956112 \[Mu]^5 + 
    1095192 \[Lambda] \[Mu]^5 + 26676 \[Lambda]^2 \[Mu]^5 - 
    4644 \[Lambda]^3 \[Mu]^5 + 1003620 \[Mu]^6 - 
    193248 \[Lambda] \[Mu]^6 - 7938 \[Lambda]^2 \[Mu]^6 + 
    1728 \[Lambda]^3 \[Mu]^6 + 1360734 \[Mu]^7 - 
    153900 \[Lambda] \[Mu]^7 - 3024 \[Lambda]^2 \[Mu]^7 + 
    216 \[Lambda]^3 \[Mu]^7 + 266358 \[Mu]^8 - 
    15030 \[Lambda] \[Mu]^8 - 324 \[Lambda]^2 \[Mu]^8 + 
    12054 \[Mu]^9 + 90 \[Lambda] \[Mu]^9 - 132 \[Mu]^10) + 
 q^4 (167930304 - 45968544 \[Lambda] - 1848216 \[Lambda]^2 + 
    426816 \[Lambda]^3 + 10584 \[Lambda]^4 + 18942024 \[Mu] - 
    9558288 \[Lambda] \[Mu] + 204228 \[Lambda]^2 \[Mu] + 
    258336 \[Lambda]^3 \[Mu] + 8316 \[Lambda]^4 \[Mu] - 
    215320070 \[Mu]^2 + 55914576 \[Lambda] \[Mu]^2 + 
    1961484 \[Lambda]^2 \[Mu]^2 - 554688 \[Lambda]^3 \[Mu]^2 - 
    10422 \[Lambda]^4 \[Mu]^2 - 25966641 \[Mu]^3 + 
    5179128 \[Lambda] \[Mu]^3 - 105708 \[Lambda]^2 \[Mu]^3 - 
    106056 \[Lambda]^3 \[Mu]^3 - 3159 \[Lambda]^4 \[Mu]^3 + 
    98017260 \[Mu]^4 - 24489756 \[Lambda] \[Mu]^4 - 
    350478 \[Lambda]^2 \[Mu]^4 + 268164 \[Lambda]^3 \[Mu]^4 + 
    2646 \[Lambda]^4 \[Mu]^4 + 12523872 \[Mu]^5 - 
    499740 \[Lambda] \[Mu]^5 - 70719 \[Lambda]^2 \[Mu]^5 - 
    24300 \[Lambda]^3 \[Mu]^5 - 945 \[Lambda]^4 \[Mu]^5 - 
    20792040 \[Mu]^6 + 5070828 \[Lambda] \[Mu]^6 + 
    4710 \[Lambda]^2 \[Mu]^6 - 39096 \[Lambda]^3 \[Mu]^6 + 
    1080 \[Lambda]^4 \[Mu]^6 - 3235641 \[Mu]^7 + 
    174492 \[Lambda] \[Mu]^7 + 22905 \[Lambda]^2 \[Mu]^7 + 
    5454 \[Lambda]^3 \[Mu]^7 - 324 \[Lambda]^4 \[Mu]^7 + 
    2162820 \[Mu]^8 - 441210 \[Lambda] \[Mu]^8 - 
    480 \[Lambda]^2 \[Mu]^8 + 1620 \[Lambda]^3 \[Mu]^8 + 
    600906 \[Mu]^9 - 52158 \[Lambda] \[Mu]^9 - 
    1563 \[Lambda]^2 \[Mu]^9 + 31536 \[Mu]^10 + 
    594 \[Lambda] \[Mu]^10 - 542 \[Mu]^11) + 
 q^3 (-501984704 + 227022336 \[Lambda] - 5248416 \[Lambda]^2 - 
    4612800 \[Lambda]^3 + 54432 \[Lambda]^4 + 12096 \[Lambda]^5 + 
    311265680 \[Mu] - 137225280 \[Lambda] \[Mu] - 
    1216272 \[Lambda]^2 \[Mu] + 2334528 \[Lambda]^3 \[Mu] + 
    62640 \[Lambda]^4 \[Mu] + 1728 \[Lambda]^5 \[Mu] + 
    632207152 \[Mu]^2 - 254520672 \[Lambda] \[Mu]^2 + 
    7974288 \[Lambda]^2 \[Mu]^2 + 5194656 \[Lambda]^3 \[Mu]^2 - 
    123768 \[Lambda]^4 \[Mu]^2 - 19008 \[Lambda]^5 \[Mu]^2 - 
    334067278 \[Mu]^3 + 148455840 \[Lambda] \[Mu]^3 - 
    3003360 \[Lambda]^2 \[Mu]^3 - 3144384 \[Lambda]^3 \[Mu]^3 - 
    13500 \[Lambda]^4 \[Mu]^3 + 4320 \[Lambda]^5 \[Mu]^3 - 
    312265260 \[Mu]^4 + 100104528 \[Lambda] \[Mu]^4 - 
    1977288 \[Lambda]^2 \[Mu]^4 - 1350756 \[Lambda]^3 \[Mu]^4 + 
    96012 \[Lambda]^4 \[Mu]^4 + 8100 \[Lambda]^5 \[Mu]^4 + 
    127718436 \[Mu]^5 - 56067408 \[Lambda] \[Mu]^5 + 
    2140092 \[Lambda]^2 \[Mu]^5 + 992556 \[Lambda]^3 \[Mu]^5 - 
    52704 \[Lambda]^4 \[Mu]^5 - 4428 \[Lambda]^5 \[Mu]^5 + 
    77361882 \[Mu]^6 - 17863248 \[Lambda] \[Mu]^6 - 
    263370 \[Lambda]^2 \[Mu]^6 + 109566 \[Lambda]^3 \[Mu]^6 + 
    3024 \[Lambda]^4 \[Mu]^6 + 486 \[Lambda]^5 \[Mu]^6 - 
    21664626 \[Mu]^7 + 9368520 \[Lambda] \[Mu]^7 - 
    375672 \[Lambda]^2 \[Mu]^7 - 119502 \[Lambda]^3 \[Mu]^7 + 
    5670 \[Lambda]^4 \[Mu]^7 + 54 \[Lambda]^5 \[Mu]^7 - 
    10392954 \[Mu]^8 + 1692648 \[Lambda] \[Mu]^8 + 
    89496 \[Lambda]^2 \[Mu]^8 + 828 \[Lambda]^3 \[Mu]^8 - 
    1350 \[Lambda]^4 \[Mu]^8 + 1503736 \[Mu]^9 - 
    633192 \[Lambda] \[Mu]^9 + 16992 \[Lambda]^2 \[Mu]^9 + 
    4380 \[Lambda]^3 \[Mu]^9 + 784106 \[Mu]^10 - 
    95490 \[Lambda] \[Mu]^10 - 3618 \[Lambda]^2 \[Mu]^10 + 
    48898 \[Mu]^11 + 1506 \[Lambda] \[Mu]^11 - 1108 \[Mu]^12) + 
 q^2 (836180096 - 547581888 \[Lambda] + 58215952 \[Lambda]^2 + 
    15863232 \[Lambda]^3 - 1449744 \[Lambda]^4 - 112320 \[Lambda]^5 + 
    3456 \[Lambda]^6 - 1126111056 \[Mu] + 762239328 \[Lambda] \[Mu] - 
    76804312 \[Lambda]^2 \[Mu] - 23801664 \[Lambda]^3 \[Mu] + 
    1515672 \[Lambda]^4 \[Mu] + 145152 \[Lambda]^5 \[Mu] - 
    1728 \[Lambda]^6 \[Mu] - 629130702 \[Mu]^2 + 
    270918872 \[Lambda] \[Mu]^2 - 19853718 \[Lambda]^2 \[Mu]^2 - 
    2485248 \[Lambda]^3 \[Mu]^2 + 1224492 \[Lambda]^4 \[Mu]^2 + 
    58752 \[Lambda]^5 \[Mu]^2 - 5184 \[Lambda]^6 \[Mu]^2 + 
    1227656497 \[Mu]^3 - 713437644 \[Lambda] \[Mu]^3 + 
    64646079 \[Lambda]^2 \[Mu]^3 + 16974480 \[Lambda]^3 \[Mu]^3 - 
    2094402 \[Lambda]^4 \[Mu]^3 - 163296 \[Lambda]^5 \[Mu]^3 + 
    4320 \[Lambda]^6 \[Mu]^3 + 129231640 \[Mu]^4 + 
    39168132 \[Lambda] \[Mu]^4 - 16005606 \[Lambda]^2 \[Mu]^4 - 
    3331044 \[Lambda]^3 \[Mu]^4 + 528570 \[Lambda]^4 \[Mu]^4 + 
    68580 \[Lambda]^5 \[Mu]^4 + 1080 \[Lambda]^6 \[Mu]^4 - 
    521227041 \[Mu]^5 + 244938894 \[Lambda] \[Mu]^5 - 
    14554761 \[Lambda]^2 \[Mu]^5 - 4373616 \[Lambda]^3 \[Mu]^5 + 
    345801 \[Lambda]^4 \[Mu]^5 + 3240 \[Lambda]^5 \[Mu]^5 - 
    2268 \[Lambda]^6 \[Mu]^5 + 961914 \[Mu]^6 - 
    40270740 \[Lambda] \[Mu]^6 + 7410078 \[Lambda]^2 \[Mu]^6 + 
    1267350 \[Lambda]^3 \[Mu]^6 - 183912 \[Lambda]^4 \[Mu]^6 - 
    4806 \[Lambda]^5 \[Mu]^6 + 864 \[Lambda]^6 \[Mu]^6 + 
    108971301 \[Mu]^7 - 38930082 \[Lambda] \[Mu]^7 + 
    577908 \[Lambda]^2 \[Mu]^7 + 477522 \[Lambda]^3 \[Mu]^7 + 
    1017 \[Lambda]^4 \[Mu]^7 - 756 \[Lambda]^5 \[Mu]^7 - 
    108 \[Lambda]^6 \[Mu]^7 - 1923114 \[Mu]^8 + 
    7368792 \[Lambda] \[Mu]^8 - 955740 \[Lambda]^2 \[Mu]^8 - 
    158694 \[Lambda]^3 \[Mu]^8 + 13848 \[Lambda]^4 \[Mu]^8 + 
    378 \[Lambda]^5 \[Mu]^8 - 11640658 \[Mu]^9 + 
    2965506 \[Lambda] \[Mu]^9 + 94459 \[Lambda]^2 \[Mu]^9 - 
    16104 \[Lambda]^3 \[Mu]^9 - 2250 \[Lambda]^4 \[Mu]^9 - 
    57450 \[Mu]^10 - 458898 \[Lambda] \[Mu]^10 + 
    35588 \[Lambda]^2 \[Mu]^10 + 5640 \[Lambda]^3 \[Mu]^10 + 
    586512 \[Mu]^11 - 97588 \[Lambda] \[Mu]^11 - 
    4332 \[Lambda]^2 \[Mu]^11 + 45172 \[Mu]^12 + 
    1854 \[Lambda] \[Mu]^12 - 1214 \[Mu]^13) + 
 q (-736463264 + 646878976 \[Lambda] - 133553952 \[Lambda]^2 - 
    17919680 \[Lambda]^3 + 5615328 \[Lambda]^4 + 215616 \[Lambda]^5 - 
    34560 \[Lambda]^6 + 1523026744 \[Mu] - 
    1397249664 \[Lambda] \[Mu] + 299842624 \[Lambda]^2 \[Mu] + 
    40965440 \[Lambda]^3 \[Mu] - 12805872 \[Lambda]^4 \[Mu] - 
    613440 \[Lambda]^5 \[Mu] + 65664 \[Lambda]^6 \[Mu] - 
    180785324 \[Mu]^2 + 427674368 \[Lambda] \[Mu]^2 - 
    152104004 \[Lambda]^2 \[Mu]^2 - 18008720 \[Lambda]^3 \[Mu]^2 + 
    7701000 \[Lambda]^4 \[Mu]^2 + 612576 \[Lambda]^5 \[Mu]^2 - 
    13824 \[Lambda]^6 \[Mu]^2 - 1420117994 \[Mu]^3 + 
    959975864 \[Lambda] \[Mu]^3 - 116928742 \[Lambda]^2 \[Mu]^3 - 
    20991552 \[Lambda]^3 \[Mu]^3 + 2856564 \[Lambda]^4 \[Mu]^3 - 
    234432 \[Lambda]^5 \[Mu]^3 - 53568 \[Lambda]^6 \[Mu]^3 + 
    618432050 \[Mu]^4 - 625781340 \[Lambda] \[Mu]^4 + 
    128365746 \[Lambda]^2 \[Mu]^4 + 17146776 \[Lambda]^3 \[Mu]^4 - 
    4455600 \[Lambda]^4 \[Mu]^4 + 34884 \[Lambda]^5 \[Mu]^4 + 
    49680 \[Lambda]^6 \[Mu]^4 + 517787598 \[Mu]^5 - 
    206340576 \[Lambda] \[Mu]^5 - 8655414 \[Lambda]^2 \[Mu]^5 + 
    3151584 \[Lambda]^3 \[Mu]^5 + 746526 \[Lambda]^4 \[Mu]^5 - 
    41148 \[Lambda]^5 \[Mu]^5 - 14040 \[Lambda]^6 \[Mu]^5 - 
    292145874 \[Mu]^6 + 218445432 \[Lambda] \[Mu]^6 - 
    26960190 \[Lambda]^2 \[Mu]^6 - 5217684 \[Lambda]^3 \[Mu]^6 + 
    685134 \[Lambda]^4 \[Mu]^6 + 31680 \[Lambda]^5 \[Mu]^6 - 
    1512 \[Lambda]^6 \[Mu]^6 - 96500154 \[Mu]^7 + 
    9734760 \[Lambda] \[Mu]^7 + 7767396 \[Lambda]^2 \[Mu]^7 + 
    300168 \[Lambda]^3 \[Mu]^7 - 272622 \[Lambda]^4 \[Mu]^7 - 
    3888 \[Lambda]^5 \[Mu]^7 + 1512 \[Lambda]^6 \[Mu]^7 + 
    59373894 \[Mu]^8 - 32704812 \[Lambda] \[Mu]^8 + 
    1650036 \[Lambda]^2 \[Mu]^8 + 623868 \[Lambda]^3 \[Mu]^8 - 
    4026 \[Lambda]^4 \[Mu]^8 - 2484 \[Lambda]^5 \[Mu]^8 - 
    216 \[Lambda]^6 \[Mu]^8 + 10244152 \[Mu]^9 + 
    1506076 \[Lambda] \[Mu]^9 - 916074 \[Lambda]^2 \[Mu]^9 - 
    94568 \[Lambda]^3 \[Mu]^9 + 14844 \[Lambda]^4 \[Mu]^9 + 
    588 \[Lambda]^5 \[Mu]^9 - 5701664 \[Mu]^10 + 
    2188272 \[Lambda] \[Mu]^10 + 28534 \[Lambda]^2 \[Mu]^10 - 
    21760 \[Lambda]^3 \[Mu]^10 - 1740 \[Lambda]^4 \[Mu]^10 - 
    639452 \[Mu]^11 - 139492 \[Lambda] \[Mu]^11 + 
    29368 \[Lambda]^2 \[Mu]^11 + 3544 \[Lambda]^3 \[Mu]^11 + 
    228448 \[Mu]^12 - 52972 \[Lambda] \[Mu]^12 - 
    2608 \[Lambda]^2 \[Mu]^12 + 23108 \[Mu]^13 + 
    1116 \[Lambda]  \[Mu]^13 - 684 \[Mu]^14) *)