The expansions of algebraic functions can be computed "fast" using the Newton Polygon Process and any "normal" aeration Let M(j) be the number of operations sufficient to multiply two jth-degree polynomials It is shown that the first N terms of an expansion of any algebraic function defined by an nth-degree polynomial can be computed in 0(nM{N)) operations, while the classical method needs O(N^n) operations Among the numerous applications of algebraic functions are symbolic mathematics and combinatorial analysis Reversion, reciprocation, and nth root of a polynomial are all special cases of algebraic functions