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.