I am considering multivariate polynomials of the form
$$f(x,y)=x^a\,y^b\,p(x,y)^c$$
(and similarly for higher dimensions). I am trying to transform these polynomials into the generic form
$$\widetilde{f}(x,y)=g_1(x)^{p_{1}}\cdots g_n(x)^{p_{n}}\,k_1(y)^{q_1}\cdots k_m(y)^{q_m}$$
via a series of coordinate transformations, with the resulting functions $g_i,k_i$ polynomials. These transformations can be anything as long as the Jacobian of the transformation is not zero.
For example, consider
$$f(x,y)=x\,y\,(s t+s (x+y)+x y)$$
with $s$, $t$ constants. By performing the transformation
$$ y\to\left(-\frac{s (t+x)}{s+x}\right)\,y \qquad x \to x $$
I obtain
$$ \widetilde{f}(x,y)=s^2 x\,(s+x)^{-1}\,(t+x)^2\,y\,(y-1) $$
which is of the desired form.
In general, whilst tackling this problem I have "discovered" that if $p(x,y)$ is linear in at least one of $x$ or $y$, (say $y$ w.l.o.g.) then the transformation
$$ y\to f(x)\,y $$
where $f(x)$ is the solution to $p(x,y)=0$ for $y$, will bring the polynomial to the desired form. This is the general case of the example illustrated above. Interestingly, to me this seems as a sort of "generalisation" of blowups, where normally one blows up a polynomial at one point, but this transformation does the trick "globally".
However, if $p(x,y)$ is quadratic in $x,y$ or worse, then the method illustrated above does not work, and I always obtain some "mixed" factors, or worse, square roots with mixed factors.
My question is, is there a known solution to this problem? Can any multivariate polynomial even be cast in such a form? If so, does anyone know of, or could point me to a solution/further reading with regards to this problem?
Thank you very much!
