  The main result provides an algorithm for determining the minimal free
resolution of ideals of fat point subschemes of ${\bf P}^2$ involving up to 8
general points with arbitrary multiplicities; the results hold over
algebraically closed fields of any characteristic. The algorithm, which works
by giving a formula in certain cases and a reduction to these cases otherwise,
does not involve Gr\"obner bases, and so is very fast, even for very large
multiplicities. Partial information is also obtained in certain cases with
$n>8$.
