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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 considers a somewhat similar problem, though restricting himself to the composition of polynomials, not homogeneous polynomials in two variables.

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 considers a somewhat similar problem, though restricting himself to the composition of univariate polynomials, not homogeneous polynomials in two variables.

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 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 considers a somewhat similar problem, though restricting himself to the composition of polynomials, not homogeneous polynomials in two variables.

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 considers a somewhat similar problem, though restricting himself to the composition of polynomials, not homogeneous polynomials in two variables.