Detropicalization as a proof technique

Rational functions make sense over any commutative ring, as long as the denominators are invertible. When there is no subtraction involved, they even apply over semirings (rings without subtraction). It is particularly worthwhile to evaluate them over the tropical semiring, in which the roles of addition and multiplication are played by maxima and addition. Over this semiring, algebraic results often acquire combinatorial meaning. We give a few examples.