Exact computations on curves
With rational Puiseux expansions you may get more than with classical Puiseux expansions for a lower price! A slight modification of Newton's polygon method speeds up the computations and yields additional invariants (the residual fields):
Dominique Duval. Rational Puiseux expansion. Compositio mathematica (70) 119-154 (1989) on NUMDAM.
Adrien Poteaux, Marc Rybowicz. Good reduction of Puiseux series and complexity of the Newton-Puiseux algorithm over finite fields. ISSAC 2008 ACM p.239-246 (2008)
The factorisation of bivariate polynomials with complex coefficients is quite "similar" to the factorisation of univariate polynomials over a finite field:
Dominique Duval. Absolute factorization of polynomials: a geometric approach. SIAM Journal on Computing (20) 1-21 (1991).