Let $P(x)$ be an integer polynomial of composite degree $D$.
A decomposition is the opposite of a composition. For instance composition of polynomials $A(x),B(x)$ gives $A(B(x)) = C(x)$. The decomposition of $C(x) = A(B(x))$.
However decomposition is not neccessarily unique. (I compare composition with multiplication in a ring and decomposition with factoring in a ring , though the analogue is not so good because ring properties are lost.)
Now I want to consider noncommutative decomposition : $A(B(x)) = C(x) =/= B(A(x)).$
The main questions are now :
1) For a given $P(x)$ how many noncommutative decompositions are there ? This could be done by " hard labour " ; testing with many dummy variables for every degree $Q$ such that $1<Q<D-1$. But Im more looking for efficient ways or theorems.
Or a fast way to know that I found all solutions.
Both the actual polynomials computations and/or just the number of solutions intrests me.
2) What are boundaries on how many noncommutative decompositions there are for a given $D$ ? For instance how many noncommutative decompositions can there be for an integer polynomial with coefficients smaller then 10^24 and of degree $D=24$ ?
A very optimist "guessed answer" would be $(d(24)-2)$ or half of that , where d is the divisors function.
I assume there are many theorems about such stuff.
Im not much intrested in irreducibility or the zero's of functions. Just saying to be clear.
Its all about these noncommutative decompositions and the amount of them.