For a given a bivariate polynomial $P(x,y)$ with rational coefficients:

Q1. How compute such (invertible) substitutions of its variables that would transform the polynomial into a homogeneous one? In particular, how to represent $P(x,x)$ as in the form $H(S(x,y),T(x,y))$, if such representation exists, where $H,S,T$ are polynomials and $H$ is homogeneous.

Q2. Can the degree of $H$ be minimized? Can maximum degree of $H,S,T$ be minimized?

Example. The polynomial $$x^6 + x^5 + (y - 6)x^4 - 3x^3 + (-y + 12)x^2 + (3y^2 + 4)x + y^3 + 4y - 8$$ can be represented as $$u^3 + uv^2 + v^3,$$ where $u=x+y$ and $v=x^2-2$.

P.S. Trivial representation with $H(u,v)=u$ and $u=P(x,y)$, $v=0$ is unacceptable as non-invertible.

The above questions and restrictions may sound too vague, so I'd be thankful if someone can help me to formalize the problem I'm trying to pose.