In the book I am reading right now, it is defined that for a map $\mu \colon \mathbb{P}^1\rightarrow \mathbb{P}^r$ the degree is the degree of the direct image cycle $\mu_{*}[\mathbb{P}^1]$. We are working over $\mathbb{C}$, so the homology of $\mathbb{P}^r$ is $H_n(\mathbb{P}^r)=\mathbb{Z}$ whenever $0\leq n\leq r$ and $n$ is even.
$[\mathbb{P}^1]$ is the fundamental class I believe, i.e. the generator of $H_2(\mathbb{P}^1)$. This is sent under the push forward to a class in $H_2(\mathbb{P}^r)$. Is then the degree just the number we get in $\mathbb{Z}$?
I wanted to explicitly calculate some degrees of specific maps. For example take $r=2$ and suppose $\mu\colon \mathbb{P}^1\rightarrow \mathbb{P}^2$ is given by a degree three homogeneous polynomials, for example $\mu(a:b)=(a^3:b^3:ab^2)$. My problem is that I don't know how to explicitly compute the degree of such a map or in general when we have degree $d$ homogeneous polynomials.