Finding mappings between expressions

Question

Suppose we have an expression of the form:

$j=\frac{A\left(t\right)}{B\left(t\right)}=\frac{C\left(s\right)}{D\left(s\right)}$

That is, $j$ can be expressed either as a function of $t$, or as a function of $s$. Is it possible to use Mathematica to find the substitution $t\rightarrow h\left(s\right)$ which takes us from the first of these forms for $j$ to the second, assuming both are known?

Actually, my own problem is slightly more complicated than this. Suppose again that we have:

$j=\frac{A\left(t\right)}{B\left(t\right)}$

And that we know this form of $j$ exactly (specifically, I am looking at the Index 36 $j$-invariants on page 5 here: http://mysite.science.uottawa.ca/asebbar/publi/mcse.pdf). This time, we want to find a substitution $t\rightarrow h\left(z_{1},z_{2}\right)$ which puts $j$ into the form:

$j=\frac{4f\left(z_{1},z_{2}\right)^{3}}{4f\left(z_{1},z_{2}\right)^{3}-27g\left(z_{1},z_{2}\right)^{2}}$

Where $f$ is a homogenous polynomial of degree 8 in each of the $z_i$, and $g$ is a homogenous polynomial of degree 12 in each of the $z_i$. Is it possible to write a Mathematica script to identify these kinds of substitutions?

Many thanks!

Edit: Here is the first $j$ from the linked document, in Mathematica input form:

j1:={(t^3+4)^3(t^3+6t^2+4)^3(t^6-6t^5+36t^4+8t^3-24t^2+16)^3}/
{t^6(t+1)^3(t^2-t+1)^3(t-2)^6(t^2+2t+4)^6}

Edit 2: Here are some explicit $j$ for the first part of my question:

j2s:={16(1+14s^4+s^8)^3}/{s^4(s^4-1)^4}

j2t:={(t^4-4t^3+8t^2+16t+16)^3(t^4+4t^3+8t^2-16t+16)^3}/
{t^4(t-2)^4(t+2)^4(t^2+4)^4}

According to page 21 here, the subsitution in this case is $s\rightarrow t/2$.

Answer 1

I don't think the second problem can be done. Here is what I tried.

Since we can normalize a coefficient to be 1 in the substitution, and given the stated degree restrictions, the substitution will be of the form t->z1+a*z2.

num = Numerator[j1 /. {t -> z1 + a*z2}];
den = Denominator[j1 /. {t -> z1 + a*z2}];

The form of g is -27*(denominator-numerator).

diff = den - num;

One think to notice is we cannot make this homogeneous. A check of the degree 33 coefficient in z1 shows that we have a term other than in z2^3.

Expand[Coefficient[diff, z1, 33]]

(* Out[402]= -695 - 7140*a^3*z2^3 *)

Anyway, we try to see if we can formulate this as a square (upon appropriate substitution for 'a'). We start by factoring it; we can work separately with each factor.

ff = Factor[diff];

Length[ff]

(* Out[427]= 3 *)

First one is just a constant so nothing to do there.

ff[[1]]

(* Out[428]= -1 *)

Second one is not too bad to work with.

ff[[2]]

(* Out[429]= 256 - 64*z1^2 + 256*z1^3 - 48*z1^5 + 960*z1^6 + 15*z1^8 + 
 232*z1^9 - z1^11 + z1^12 - 
   128*a*z1*z2 + 768*a*z1^2*z2 - 240*a*z1^4*z2 + 5760*a*z1^5*z2 + 
 120*a*z1^7*z2 + 
   2088*a*z1^8*z2 - 11*a*z1^10*z2 + 12*a*z1^11*z2 - 64*a^2*z2^2 + 
 768*a^2*z1*z2^2 - 
   480*a^2*z1^3*z2^2 + 14400*a^2*z1^4*z2^2 + 420*a^2*z1^6*z2^2 + 
 8352*a^2*z1^7*z2^2 - 
   55*a^2*z1^9*z2^2 + 66*a^2*z1^10*z2^2 + 256*a^3*z2^3 - 
 480*a^3*z1^2*z2^3 + 
   19200*a^3*z1^3*z2^3 + 840*a^3*z1^5*z2^3 + 19488*a^3*z1^6*z2^3 - 
   165*a^3*z1^8*z2^3 + 220*a^3*z1^9*z2^3 - 240*a^4*z1*z2^4 + 
 14400*a^4*z1^2*z2^4 + 
   1050*a^4*z1^4*z2^4 + 29232*a^4*z1^5*z2^4 - 330*a^4*z1^7*z2^4 + 
 495*a^4*z1^8*z2^4 - 
   48*a^5*z2^5 + 5760*a^5*z1*z2^5 + 840*a^5*z1^3*z2^5 + 
 29232*a^5*z1^4*z2^5 - 
   462*a^5*z1^6*z2^5 + 792*a^5*z1^7*z2^5 + 960*a^6*z2^6 + 
 420*a^6*z1^2*z2^6 + 
   19488*a^6*z1^3*z2^6 - 462*a^6*z1^5*z2^6 + 924*a^6*z1^6*z2^6 + 
 120*a^7*z1*z2^7 + 
   8352*a^7*z1^2*z2^7 - 330*a^7*z1^4*z2^7 + 792*a^7*z1^5*z2^7 + 
 15*a^8*z2^8 + 
   2088*a^8*z1*z2^8 - 165*a^8*z1^3*z2^8 + 495*a^8*z1^4*z2^8 + 
 232*a^9*z2^9 - 
   55*a^9*z1^2*z2^9 + 220*a^9*z1^3*z2^9 - 11*a^10*z1*z2^10 + 
 66*a^10*z1^2*z2^10 - 
   a^11*z2^11 + 12*a^11*z1*z2^11 + a^12*z2^12 *)

Here we take the resultant of this factor with its derivative with respect to z1. If there is a nontrivial square (for some value(s) of 'a') then that will manifest as a resultant polynomial in {z2,a} (or so he said).

rr = Resultant[ff[[2]], D[ff[[2]], z1], z1]

(* Out[430]= \
-1445873207846766167649299735454865070627269701494463352799232 *)

It's constant, so no hope for proceeding further. That said, I may have misunderstood the question or made a misstep somewhere in the analysis.