Factorization of a polynomial fraction to a certain combinations of other polynomial fractions

Question

I currently am working on a project for to find some new algebra structures. But I faced with a factorization problem which I cannot come out of it! I could find the lower dimensional of it. But I need a Mathematica codding algorithm for to find the higher dimensional cases. One of my easiest factorization is as what I explain bellow (Which I know its factorization, but I want to re-find it by using Mathematica!):

I need to factorize the following polynomial fraction by using Mathematica

F6[x1, y1, x2 , y2, x3 , y3, x4, y4, x5, y5, x6] = (2 x1 x2 x5 x6 y2 (x2 y1 + x3 (y1 + y2)) y3^2 y4 (x5 y5 +  x4 (y4 + y5)))/((x2 y2 + x1 (y1 + y2)) (x3 y3+ x2 (y2 + y3))^2 (x4 y3 + x5 (y3 + y4))^2 (x5 y4 + x6 (y4 + y5)));

to the combination of the following polynomial fractions

K1[x1, y1, x2, y2, x3, y3] = ((x2 y2 + x1 (y1 + y2)) (x3 y3 + x2 (y2 + y3)))/(x2 y2 (x3 y3 + x2 (y2 + y3) + x1 (y1 + y2 + y3)));


K2[y1, x2, y2, x3, y3, x4] = ((x2 y1 + x3 (y1 + y2)) (x3 y2 + x4 (y2 + y3)))/(x3 y2 (x2 y1 + (x3 + x4) (y1 + y2) + x4 y3));


K3[x2, y2, x3, y3, x4, y4] = ((x3 y3 + x2 (y2 + y3)) (x4 y4 + x3 (y3 + y4)))/(x3 y3 (x4 y4 + x3 (y3 + y4) + x2 (y2 + y3 + y4)));


K4[ y2, x3, y3, x4, y4, x5]  = ((x3 y2 + x4 (y2 + y3)) (x4 y3 + x5 (y3 + y4)))/(x4 y3 (x3 y2 + (x4 + x5) (y2 + y3) + x5 y4));


K5[ x3, y3, x4, y4, x5, y5]  = ((x4 y4 + x3 (y3 + y4)) (x5 y5 + x4 (y4 + y5)))/(x4 y4 (x5 y5 + x4 (y4 + y5) + x3 (y3 + y4 + y5)));


K6[ y3, x4, y4, x5, y5, x6]  = ((x4 y3 + x5 (y3 + y4)) (x5 y4 + x6 (y4 + y5)))/(x5 y4 (x4 y3 + (x5 + x6) (y3 + y4) + x6 y5));

But as I know the answer, so I will write it here and need a codding which can give me this factorization or any other factorization with the above mentioned factors:

The combination is as follows:

F6[x1, y1, x2 , y2, x3 , y3, x4, y4, x5, y5, x6] = ((K1[x1, y1, x2, y2, x3, y3] - 1) (K3[x2, y2, x3, y3, x4, y4] - 1) (K4[y2, x3, y3, x4, y4, x5] - 1)(K6[y3, x4, y4, x5, y5, x6] - 1))/(K1[x1, y1, x2, y2, x3, y3] K3[x2, y2, x3, y3, x4, y4] K4[y2, x3, y3, x4, y4, x5] K6[y3, x4, y4, x5, y5, x6]) = ((K1[x1, y1, x2, y2, x3, y3] - 1)/K1[x1, y1, x2, y2, x3, y3]) ((K3[x2, y2, x3, y3, x4, y4] - 1)/K3[x2, y2, x3, y3, x4, y4]) ((K4[y2, x3, y3, x4, y4, x5] - 1)/K4[y2, x3, y3, x4, y4, x5]) ((K6[y3, x4, y4, x5, y5, x6] - 1)/K6[y3, x4, y4, x5, y5, x6]);

I will appreciate a lot if someone helps me to write down a Mathematica codding which is able to give me such combinations of $K1, K2, K3, K4, K5 $ and $K6$.

Here there is another factorization problem in 9 variable (The easiest one!) which I still don't know how it should done!

The question is the same as previous one; We have the following polynomial fraction and wanna to decompose it

F9[x1, y1, z1, x2 , y2, z2, x3 , y3, z3, x4, y4, z4, x5, y5, z5, x6, y6, z6] = (2 x1 x2 x5 y2 y5 y6 z2 (x2 y1 y2 z1 + x2 y1 y3 z1 + x3 y1 y3 z1 + x2 y1 y3 z2 + x3 y1 y3 z2 + x3 y2 y3 z2) z3^2 z4 (x4 x5 y4 z4 + x4 x6 y4 z4 + x4 x6 y5 z4 + x4 x6 y4 z5 + x4 x6 y5 z5 +  x5 x6 y5 z5))/((x1 y1 z1 + x1 y1 z2 + x1 y2 z2 + x2 y2 z2) (x2 y2 z2 + x2 y2 z3 + x2 y3 z3 +  x3 y3 z3)^2 (x4 y4 z3 + x4 y5 z3 + x5 y5 z3 + x5 y5 z4)^2 (x5 y5 z4 + x5 y6 z4 + x6 y6 z4 + x6 y6 z5));

to the combination of the following polynomial fractions

K1[x1, y1, z1, x2, y2, z2, x3, y3, z3]  = ((x1 y1 z1 + x2 y2 z2 + x1 (y1 + y2) z2) (x2 y2 z2 + x3 y3 z3 + x2 (y2 + y3) z3))/(x2 y2 z2 (x2 y2 z2 + x3 y3 z3 + x2 (y2 + y3) z3 + x1 (y2 z2 + (y2 + y3) z3 + y1 (z1 + z2 + z3))));

K2[y1, z1, x2, y2, z2, x3, y3, z3, x4] = (((x2 + x3) y1 z1 + x3 (y1 + y2) z2) ((x3 + x4) y2 z2 +  x4 (y2 + y3) z3))/(x3 y2 z2 (x2 y1 z1 + (x3 + x4) (y2 z2 + y1 (z1 + z2)) + x4 (y1 + y2 + y3) z3));


K3[z1, x2, y2, z2, x3, y3, z3, x4, y4] = ((x2 (y2 + y3) z1 + x3 y3 (z1 + z2)) (x3 (y3 + y4) z2 +  x4 y4 (z2 + z3)))/(x3 y3 z2 (x2 (y2 + y3 + y4) z1 + x3 (y3 + y4) (z1 + z2) + x4 y4 (z1 + z2 + z3)));


K4[x2, y2, z2, x3, y3, z3, x4, y4, z4]  = ((x2 y2 z2 + x3 y3 z3 + x2 (y2 + y3) z3) (x3 y3 z3 + x4 y4 z4 + x3 (y3 + y4) z4))/(x3 y3 z3 (x3 y3 z3 + x4 y4 z4 + x3 (y3 + y4) z4 + x2 (y3 z3 + (y3 + y4) z4 + y2 (z2 + z3 + z4))));


K5[y2, z2, x3, y3, z3, x4, y4, z4, x5]  = (((x3 + x4) y2 z2 + x4 (y2 + y3) z3) ((x4 + x5) y3 z3 + x5 (y3 + y4) z4))/(x4 y3 z3 (x3 y2 z2 + (x4 + x5) (y3 z3 +  y2 (z2 + z3)) + x5 (y2 + y3 + y4) z4));


K6[z2, x3, y3, z3, x4, y4, z4, x5, y5]  = ((x3 (y3 + y4) z2 + x4 y4 (z2 + z3)) (x4 (y4 + y5) z3 + x5 y5 (z3 + z4)))/(x4 y4 z3 (x3 (y3 + y4 + y5) z2 +  x4 (y4 + y5) (z2 + z3) + x5 y5 (z2 + z3 + z4)));


K7[x3, y3, z3, x4, y4, z4, x5, y5, z5]  = ((x3 y3 z3 + x4 y4 z4 + x3 (y3 + y4) z4) (x4 y4 z4 + x5 y5 z5 + x4 (y4 + y5) z5))/(x4 y4 z4 (x4 y4 z4 + x5 y5 z5 + x4 (y4 + y5) z5 + x3 (y4 z4 + (y4 + y5) z5 + y3 (z3 + z4 + z5))));


K8[y3, z3, x4, y4, z4, x5, y5, z5, x6]  = (((x4 + x5) y3 z3 + x5 (y3 + y4) z4) ((x5 + x6) y4 z4 + x6 (y4 + y5) z5))/(x5 y4 z4 (x4 y3 z3 + (x5 + x6) (y4 z4 + y3 (z3 + z4)) + x6 (y3 + y4 + y5) z5));



K9[z3, x4, y4, z4, x5, y5, z5, x6, y6]  = ((x4 (y4 + y5) z3 + x5 y5 (z3 + z4)) (x5 (y5 + y6) z4 +  x6 y6 (z4 + z5)))/(x5 y5 z4 (x4 (y4 + y5 + y6) z3 + x5 (y5 + y6) (z3 + z4) + x6 y6 (z3 + z4 + z5)));

The question is the same as the previous one. We want to decompose $F9$ just in terms of $K1, K2, K3, K4, K5, K6, K7, K8, K9$, which will be our factors.

Please again let me know if the question is not clear.

Thank you very much for your help!

Answer 1

A generalization of the answer to 129970 solves the first half of this question in about 70 seconds.

tp = Tuples[Range[-1, 1], 6]; tp // Length
(* 729 *)
gp = Map[t1^#[[1]] t2^#[[2]] t3^#[[3]] t4^#[[4]] t5^#[[5]] t6^#[[6]] &, tp].
    Table[Unique["c"], tp // Length];
sol = Flatten@Solve[Table[(f6 == (gp /. {t1 -> k1, t2 -> k2, t3 -> k3, t4 -> k4, 
    t5 -> k5, t6 -> k6})) /. Thread[{x1, x2, x3, x4, x5, x6, y1, y2, y3, y4, y5} -> 
    RandomInteger[{1, 11}, 11]], {n, tp // Length}], 
    List @@ (First@# & /@ (gp /. gp[[1]] -> gp[[1]] z))]);
sol /. Rule[_, 0] -> Nothing
(* {c114 -> -2, c115 -> 2, c123 -> 2, c124 -> -2, c330 -> -2, c331 -> 2, 
    c339 -> 2, c340 -> -2, c357 -> 2, c358 -> -2, c366 -> -2, c367 -> 2, 
    c87 -> 2, c88 -> -2, c96 -> -2, c97 -> 2} *)
Factor[gp /. sol]
(* (2 (-1 + t1) (-1 + t3) (-1 + t4) (-1 + t6))/(t1 t3 t4 t6) *)

Answer 2

Rewriting your definitions slightly

poly = (2 x1 x2 x5 x6 y2 (x2 y1 + x3 (y1 + y2)) y3^2 y4 (x5 y5 + 
   x4 (y4 + y5)))/((x2 y2 + x1 (y1 + y2)) (x3 y3 + x2 (y2 + y3))^2 (x4 y3 + 
   x5 (y3 + y4))^2 (x5 y4 + x6 (y4 + y5)));

eqns = { 
   K1 == ((x2 y2 + x1 (y1 + y2)) (x3 y3 + x2 (y2 + y3)))/(x2 y2 (x3 y3 + 
     x2 (y2 + y3) + x1 (y1 + y2 + y3))), 
   K2 == ((x2 y1 + x3 (y1 + y2)) (x3 y2 + 
     x4 (y2 + y3)))/(x3 y2 (x2 y1 + (x3 + x4) (y1 + y2) + x4 y3)), 
   K3 == ((x3 y3 + x2 (y2 + y3)) (x4 y4 + x3 (y3 + y4)))/(x3 y3 (x4 y4 + 
     x3 (y3 + y4) + x2 (y2 + y3 + y4))), 
   K4 == ((x3 y2 + x4 (y2 + y3)) (x4 y3 + 
     x5 (y3 + y4)))/(x4 y3 (x3 y2 + (x4 + x5) (y2 + y3) + x5 y4)), 
   K5 == ((x4 y4 + x3 (y3 + y4)) (x5 y5 + x4 (y4 + y5)))/(x4 y4 (x5 y5 + 
     x4 (y4 + y5) + x3 (y3 + y4 + y5))), 
   K6 == ((x4 y3 + x5 (y3 + y4)) (x5 y4 + 
     x6 (y4 + y5)))/(x5 y4 (x4 y3 + (x5 + x6) (y3 + y4) + x6 y5))
   };

Compute GroebnerBasis

gb = GroebnerBasis[eqns, {x1, y1, x2, y2, x3, y3, x4, y4, x5, y5, x6}];

The remainder gives a representation of poly in terms of K1 .. K6

{qs, r} = PolynomialReduce[poly, gb, {x1, y1, x2, y2, x3, y3, x4, y4, x5, y5, x6}];

Where r is your solution in K1 ... K6. This validates correctness:

poly == r /. ToRules[And @@ eqns] // Expand

Please note, this may take a while, but simplification seems part of your research objective (and you are only asking for ways to code this).