  As one of the first applications of the polynomial method in combinatorics,
Alon and Tarsi gave a way to prove that a graph is choosable (colorable from
any lists of prescribed size). We describe an efficient way to implement this
approach, making it feasible to test choosability of graphs with around 70
edges. We also show that in case that Alon-Tarsi method fails to show that the
graph is choosable, further coefficients of the graph polynomial provide
constraints on the list assignments from which the graph cannot be colored.
This often enables us to confirm colorability from a given list assignment, or
to decide choosability by testing just a few list assignments. The
implementation can be found at
https://gitlab.mff.cuni.cz/dvorz9am/alon-tarsi-method.
