  There are many viewpoints on algebraic power series, ranging from the
abstract ring-theoretic notion of Henselization to the very explicit
perspective as diagonals of certain rational functions. To be more explicit on
the latter, Denef and Lipshitz proved in 1987 that any algebraic power series
in $n$ variables can be written as a diagonal of a rational power series in one
variable more. Their proof uses a lot of involved theory and machinery which
remains hidden to the reader in the original article. In the present work we
shall take a glimpse on these tools by motivating while defining them and
reproving most of their interesting parts. Moreover, in the last section we
provide a new significant improvement on the Artin-Mazur lemma, proving the
existence of a 2-dimensional code of algebraic power series.
