  In this paper we address the problem of computing $\text{deg}(f^n)$, the
degrees of iterates of a birational map
$f:\mathbb{P}^N\rightarrow\mathbb{P}^N$. For this goal, we develop a method
based on two main ingredients: the factorization of a polynomial under
pull-back of $f$, based on local indices of a polynomial associated to blow-ups
used to resolve the contraction of hypersurfaces by $f$, and the propagation of
these indices along orbits of $f$.
  For maps admitting algebraically stable modifications $f_X:X\rightarrow X$,
where $X$ is a variety obtained from $\mathbb P^N$ by a finite number of
blow-ups, this method leads to an algorithm producing a finite system of
recurrent equations relating the degrees and indices of iterated pull-backs of
linear polynomials. We illustrate the method by three representative
two-dimensional examples. It is actually applicable in any dimension, and we
will provide a number of three-dimensional examples as a separate companion
paper.
