Image of a hypersurface under a map $\mathbb CP^n\to \mathbb CP^n$ Let $H$ be a degree $d$ hypersurface in $\mathbb CP^n$ defined by an explicit equation $F=0$. Let $\varphi: \mathbb P^n \to \mathbb P^n$ be an explicit degree $m$ morphism. In this case $\varphi(H)$ is a degree $d^{n-1}m$ hypersurface. Is there an algorithm to calculate the coefficients of the degree $d^{n-1}m$ polynomial that defines this hypersuface?
 A: As suggested by Alex in his comment, this is a classical problem involving  elimination theory.
The relevant result is the following, see [Greuel - Pfister, A Singular introduction to commutative algebra, Proposition A.7.12 p. 505].

Proposition. Let $f=(f_0: \ldots :f_m)\colon \mathbb{P}^n \longrightarrow \mathbb{P}^m$ be a morphism, with $f_i \in \mathbb{K}[x_0, \ldots, x_n]$ homogeneous polynomials of the same degree without common zeroes. Moreover, let $I \subset \mathbb{K}[x_0, \ldots, x_n]$ be a homogeneous ideal, defining the projective variety $X=V(I) \subset \mathbb{P}^n$, and set $$J= \langle I, \, f_0-y_0, \ldots, f_m-y_m \rangle \cap \mathbb{K}[y_0, \ldots, y_m].$$ Then $$f(X)=V(J)\subset \mathbb{P}^m.$$ 

This can be explicitly determined by using Groebner bases. The computation is usually unpractical by hands, but it can be  done with all the most common Computer Algebra Systems.
For instance, many explicit computations with $\textrm{Singular}$ can be found in the book by Greuel and Pfister quoted above.
