Equivalent binary forms 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 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.  
 A: It is true that minimal generators for algebras of invariants or covariants of binary forms are not known except for small degree $d$. However, there are plenty of covariants that are known or can be constructed, for any degree, and that should help in separating orbits.
To use a metaphor related to current events, deciding to separate orbits without using covariants would be a bit like Germany deciding to take on Brazil without using a goalkeeper.
Anyway, here are two ideas which may help in your search for an efficient algorithm that distinguishes $SL_2$ orbits of binary forms.


*

*You could use the signature curves described in Chapter 8 of the book "Classical Invariant Theory" by Olver.
Given a binary form $F$, consider the following covariants, written using transvectants:
the hessian $H=(F,F)_2$, the cubicovariant $T=(F,H)_1$ and the degree four covariant $U=(F,T)_1$. The signature curve in $\mathbb{C}^2$ is
$$
S=\left\{
\left( \frac{T(p,1)^2}{H(p,1)^3},\frac{U(p,1)}{H(p,1)^2}
\right)\ {\rm for}\ p\in\mathbb{C}\ {\rm such\ that}\ H(p,1)\neq 0
\right\}\ .
$$
Theorem 8.61 in Olver's book says that two nondegenerate binary forms are equivalent iff their signature curves are equal. Here, nondegenerate means the Hessian is not identically zero, i.e., the form is not a power of a linear form. I suppose one could even visually separate two given inequivalent binary forms by plotting suitable real slices of their signature curves. 

*You could try to find enough covariants to separate orbits. There has been some activity on finding such separating sets of invariants/covariants. See in particular this paper by Elmer and Kohls. The remarks at the bottom of page 137 of this article seems to indicate that for binary forms the problem is not completely solved yet. Yet it seems more tractable to try to find separating rather than generating sets of covariants.
Finally the answers to MO question Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$  may also help.
A: The separating set coincides with the field of semi-invariants and the last  can be easy computed in an explicit way for any degree $d.$
Precisely, it generated by  elements
$$
a_0,z_2,z_3,\ldots,z_d,
$$
where
$$
z_i:= \sum_{k=0}^{i-2} (-1)^k {i \choose k} a_{i-k}  a_1^k a_0^{i-k-1} +(i-1)(-1)^{i+1} a_1^i, i=2,\ldots,d,
$$
and  $a_i$  are the coefficients of a binary form 
$$
\sum_{i=0}^d a_i {d \choose i} x^{d-i} y^i.
$$
So, you need take two binary forms with equal $a_0$ and just calculate and compare all $z_i$.
Am  I  wrong?
A: 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, Michael Stoll, 
Stoll, Michael; Cremona, John E. On the reduction theory of binary forms. J. Reine Angew. Math. 565 (2003), 79–99. 
