I want to find a second-order approximation to $z(x,y)$ for $x$ and $y$ close to $0$. The relation between them is given by $P(x,y,z)=0$, where
$P(x,y,z)=Q(z) + R(x,y,z)$, where
$Q(z)=a_0+a_1z+a_2z^2+a_3z^3+a_4z^4+a_5z^5$ and
$R(x,y,z)=(x+1)\cdot y\cdot(b_0+b_1z+b_2z^2+b_3z^3+b_4z^4+b_5z^5)$
I tried to apply Newton's method for approximating algebraic curves with Puiseux series, but I couldn't find a good description of a general enough version of this method/algorithm in simple enough terms for me. I only managed to find a zeroth-order approximation.
What specific procedure/algorithm could be used to solve this problem?
P.S.: Please don't downvote the question before giving me a chance to improve it, I'm sure it's not perfect, and am very willing to improve/fix it if told how to do so.