I am currently reading Arkani-Hamed et al. on Positive Geometries and Canonical forms (https://arxiv.org/abs/1703.04541).
There, they define the pushforward of a canonical form by
The canonical forms of different pseudo-positive geometries can be related by certain maps between them. We begin by defining the push-forward (often also called the trace map) for differential forms, see [10, II(b)]. Consider a surjective meromorphic map φ : M → N between complex manifolds of the same dimension. Let ω be a meromorphic top form on M, b a point in N, and V an open subset containing b. If the map φ is of degree deg φ, then the pre-image $\varphi^{-1}(V)$ is the union of disconnected open subsets $U_i$ for i = 1, . . . , deg φ, with $a_i \in U_i$ and $φ(a_i) = b$.
We define the push-forward as a meromorphic top form on N in the following way: $$φ_*(ω)(b) := \Sigma_i ψ^∗_i(ω(a_i))$$ where $ψ_i:= φ|_{U_i}^{-1}: V → U_i$.
In other words, they are summing the pullbacks of the inverse function over all of the roots.
Whilst this idea is quite clear, I don't get how to do epxlicit calculations: In example 25 in the same paper, they calculate the canonical form of a "pizza slice" by pushforward.
The corresponding map is given by $\varphi(z,t) = (1, \frac{1-z^2}{(1+z^2)(1+t)}, \frac{2z}{(1+z^2)(1+t)})$.
They claim, the pushforward of the canonical form $\frac{(z_2-z_1)}{(z_2-z)(z-z_1)}dz\frac{1}{t}dt$ under this map $\varphi$ is given by
$$\frac{(1+t)^3}{2}(\frac{(1+z^2)(z_2-z_1)}{(z_2-z)(z-z_1)t}-\frac{(1+z^2)(z_2-z_1)}{(zz_2+1)(1+zz_1)(t+2)})dxdy$$
I get why there are two roots and they are related by the following relation: if $(z,t)$ is the first root, then $(-1/z, -t-2)$ is the other one.
However, I don't get how to explicitly caclulate the pullback of the inverse function. Any help is appreciated!