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.