Let $F(x,y), A(x,y), B(x,y)$ be homogeneous polynomials with integer coefficients (a.k.a. binary forms) such that the degrees of $A(x,y)$ and $B(x,y)$ match. Define


$$R(x,y) := F\left(A(x,y), B(x,y)\right).$$


Then $R(x,y)$ is a homogeneous polynomial with integer coefficients of degree $dr$, where $d := \deg(F)$ and $r := \deg(A) = \deg(B)$. We identify the discriminant $D(R)$ of $R(x,y)$ with the discriminant of the polynomial $R(x,1)$. Similarly, we define the discriminants of $F, A, B$.


I would like to obtain the formula for $D(R)$. Alternatively, I need to bound $D(R)$ from below non-trivially in terms of the invariants of $F, A, B$, such as their discriminants. Has this question been explored at all?


It seems to me that John Cullinan in his short article [*The discriminant of a composition of two polynomials*](http://math.bard.edu/cullinan/expository/disccomposite.pdf) considers a somewhat similar problem, though restricting himself to the composition of univariate polynomials, not homogeneous polynomials in two variables.



Discriminant of a composition of binary forms
If I did not mess up the adaptation to your special case, Theorem 3.31 for $n=2$ in ["A computational approach to the discriminant of homogeneous polynomials"](https://arxiv.org/abs/1210.4697) by Busé and Jouanolou
says that
$$
{\rm Disc}(R)=({\rm Disc}(F))^r\times
{\rm Res}(A,B)^{d(d-1)}\times K(F,A,B)
$$
where $K$ is homogeneous of degree $2d(r-1)$ in the coefficients of $A$ or $B$ and of degree $2(r-1)$ in the coefficients of $F$.


