a classical result of skolem, mahler, and lech asserts that a linearly recurrent sequence taking values in a field of characteristic zero has the property that its zero set is a finite union of one-way infinite arithmetic progressions along with a finite set. in positive characteristic, examples due to lech show that this conclusion does not hold. for years the problem of finding a positive characteristic analogue was open until it was solved by derksen in 2005. we describe extensions of derksen's work involving finite-state machines and explain how these extensions allow one to effectively solve many classes of diophantine problems in positive characteristic.