3
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Consider the large expression LE defined in this file (involving only linearly independent functions), or in this file (linear interdependence present but slightly shorter notation). Expression LE contains variables q[i] with $i=1,2,3,4$ and coefficients consisting of f[x, y] terms with $x,y\in\mathbb{N}^+$. Overall, the expression LE above features each term appearing in the following list multiplied by some coefficient dependent on the f[x, y]:

vL = {q[1]^6, q[1]^5 q[2], q[1]^4 q[2]^2, q[1]^3 q[2]^3, q[1]^2 q[2]^4, q[1] q[2]^5, q[2]^6, q[1]^5 q[3], q[1]^4 q[2] q[3], q[1]^3 q[2]^2 q[3], q[1]^2 q[2]^3 q[3], q[1] q[2]^4 q[3], q[2]^5 q[3], q[1]^4 q[3]^2, q[1]^3 q[2] q[3]^2, q[1]^2 q[2]^2 q[3]^2, q[1] q[2]^3 q[3]^2, q[2]^4 q[3]^2, q[1]^3 q[3]^3, q[1]^2 q[2] q[3]^3, q[1] q[2]^2 q[3]^3, q[2]^3 q[3]^3, q[1]^2 q[3]^4, q[1] q[2] q[3]^4, q[2]^2 q[3]^4, q[1] q[3]^5, q[2] q[3]^5, q[3]^6, q[1]^5 q[4], q[1]^4 q[2] q[4], q[1]^3 q[2]^2 q[4], q[1]^2 q[2]^3 q[4], q[1] q[2]^4 q[4], q[2]^5 q[4], q[1]^4 q[3] q[4], q[1]^3 q[2] q[3] q[4], q[1]^2 q[2]^2 q[3] q[4], q[1] q[2]^3 q[3] q[4], q[2]^4 q[3] q[4], q[1]^3 q[3]^2 q[4], q[1]^2 q[2] q[3]^2 q[4], q[1] q[2]^2 q[3]^2 q[4], q[2]^3 q[3]^2 q[4], q[1]^2 q[3]^3 q[4], q[1] q[2] q[3]^3 q[4], q[2]^2 q[3]^3 q[4], q[1] q[3]^4 q[4], q[2] q[3]^4 q[4], q[3]^5 q[4], q[1]^4 q[4]^2, q[1]^3 q[2] q[4]^2, q[1]^2 q[2]^2 q[4]^2, q[1] q[2]^3 q[4]^2, q[2]^4 q[4]^2, q[1]^3 q[3] q[4]^2, q[1]^2 q[2] q[3] q[4]^2, q[1] q[2]^2 q[3] q[4]^2, q[2]^3 q[3] q[4]^2, q[1]^2 q[3]^2 q[4]^2, q[1] q[2] q[3]^2 q[4]^2, q[2]^2 q[3]^2 q[4]^2, q[1] q[3]^3 q[4]^2, q[2] q[3]^3 q[4]^2, q[3]^4 q[4]^2, q[1]^3 q[4]^3, q[1]^2 q[2] q[4]^3, q[1] q[2]^2 q[4]^3, q[2]^3 q[4]^3, q[1]^2 q[3] q[4]^3, q[1] q[2] q[3] q[4]^3, q[2]^2 q[3] q[4]^3, q[1] q[3]^2 q[4]^3, q[2] q[3]^2 q[4]^3, q[3]^3 q[4]^3, q[1]^2 q[4]^4, q[1] q[2] q[4]^4, q[2]^2 q[4]^4, q[1] q[3] q[4]^4, q[2] q[3] q[4]^4, q[3]^2 q[4]^4, q[1] q[4]^5, q[2] q[4]^5, q[3] q[4]^5, q[4]^6};

Since LE is the U-resultant of a system of multivariate equations, it is mathematically guaranteed to factor in the following fashion:

myFactoring = Product[Sum[C[j, i] q[j], {j, 1, 4}], {i, 1, 6}];

With coefficients of interest C[j, i] which will then be used to construct solutions to the multivariate system of equations. Expanding myFactoring we see that each variable combination from vL properly appears. Unfortunately, LE is so large and unwieldy that I do not know how to go about to find the explicit factorization. So far I could find out by other means that square roots will appear in the factorization. The C[j, i] might be complex depending on what the parameters f[i, j] are and always two out of the six factorized brackets will be complex conjugates of each other (assuming q[i] are real). Any suggestion on how to do it?

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  • $\begingroup$ Is LE the Det of 91859? $\endgroup$ – bbgodfrey Aug 20 '15 at 3:35
  • $\begingroup$ It is related to it. Multiplying the Det by a constant factor and dividing out a different determinant (basically also a constant) gives LE. So, yes, one could also factorize the Det keeping in mind that an extra overall factor dependent on the f[x,y] should appear then. $\endgroup$ – Kagaratsch Aug 20 '15 at 6:16
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    $\begingroup$ It may be useful to know that q[i] appears only in the combination q[i] - (f[i, 6] q[1])/f[1, 6] with {i, 2, 4}. This can be seen by examining rr from the previous question. $\endgroup$ – bbgodfrey Aug 20 '15 at 12:58
  • $\begingroup$ Note also that Sum[C[j, i] q[j], {j, 1, 4}] can be scaled by an arbitrary factor p[i], constrained only by Product[p[i], {i, 6}] == 1. Does this matter? $\endgroup$ – bbgodfrey Aug 21 '15 at 0:07
  • $\begingroup$ Since the relevant solutions at the end of the day are given by C[1,i]/C[4,i] and C[2,i]/C[4,i] and C[3,i]/C[4,i] any such scaling would drop out. I am not sure if such scaling might be useful at intermediate steps though. $\endgroup$ – Kagaratsch Aug 21 '15 at 0:33

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