Quote:
Originally Posted by grandpascorpion
Thank you both for your feedback.
I wonder if a better tack would be to check if one (or more) of these polynomials (in three variables: n,d and k) can be factored into two smaller polynomials say g(n,d,k) and h(n,d,k).

This is trivial with magma:
Code:
P<n,d,k>:=PolynomialRing(Rationals(),3);
F:=(4*n^6  4*d*n^5  3*d^2*n^4 + 6*d^3*n^3  4*d^4*n^2)*k^4 +
(8*n^7 + 8*d*n^6  26*d^2*n^5 + 10*d^3*n^4 + 6*d^4*n^3  12*d^5*n^2)*k^3 +
(4*n^8 + 28*d*n^7  23*d^2*n^6  32*d^3*n^5 + 42*d^4*n^4  24*d^5*n^3  8*d^6*n^2)*k^2 +
(16*d*n^8+ 16*d^2*n^7  52*d^3*n^6 + 20*d^4*n^5 + 12*d^5*n^4  24*d^6*n^3)*k +
(16*d^2*n^8  16*d^3*n^7  12*d^4*n^6 + 24*d^5*n^5  16*d^6*n^4);
Factorisation(F);
but the factors are uninteresting:
Code:
[
<d + 1/2*k, 1>,
<n, 2>,
<n + k, 1>,
<n^5*d + 1/2*n^5*k  n^4*d^2 + 1/2*n^4*d*k + 1/2*n^4*k^2  3/4*n^3*d^3 
15/8*n^3*d^2*k  1/2*n^3*d*k^2 + 3/2*n^2*d^4 + 5/4*n^2*d^3*k 
3/8*n^2*d^2*k^2  n*d^5  1/4*n*d^4*k + 3/4*n*d^3*k^2  1/2*d^5*k 
1/2*d^4*k^2, 1>
]