Two binary forms $f, g \in k[x, y]$ are equivalent when there exists an $M \in GL\_2 (k)$ such that $f^M = g$. For simplicity we take $k$ such that $char (k) =0$ and $k=\bar k$. 


The equivalence classes of binary forms are determined by the $GL\_2 (k)$-invariants and they are known for degree $d \leq 8$ (and possibly $d=9, 10$). Hence, $f$ is equivalent to $g$ if and only if they have the same invariants.


Is there a more efficient way of doing this? Has anybody tried to implement any algorithm to check $GL\_2 (k)$-equivalence for binary forms of degree $d > 8$ or for any group $G \leq GL\_2 (k)$?


Example: Let $f(x, z)$ be the binary form given by $$f(x, 1)= 442765625 x^6-719030400000 x^5+320847859200000 x^4-64095440076800000 x^3+6360693303410688000 x^2-282590704159256739840 x+3449767488965367037952$$
A simple Maple program by Mark van Hoeji will show that this is equivalent to
$$g(x, 1)=28337 x^6-326832 x^5+1035795 x^4-1469600 x^3+1035795 x^2-326832 x+28337$$
Indeed, I know by checking the $GL\_2 (\mathbb C)$-invariants $i\_1, i\_2, i\_3$ (see for example this [paper](http://arxiv.org/abs/1209.0439) among others for their definitions) that $f$ and $g$ are equivalent to $$h(x, 1)= x^5+x^4+x^3+x^2+x.$$


So, my question is if there is any simple approach that would work without using the whole machinery of invariant theory. The main reason that I would like to avoid invariants is that for high degrees we don't know them explicitly. 



Equivalent binary forms
From what I have found it seems as the best way to do this is to use an algorithm of Stoll on reducing the binary form. The algorithm is based on Julia's invariant and it is very nice. 


For details please see


[Reduction theory of point clusters in projective space](http://arxiv.org/abs/0909.2808), Michael Stoll, 


Stoll, Michael; Cremona, John E. On the reduction theory of binary forms. J. Reine Angew. Math. 565 (2003), 79–99. 


