We show that for several classes of idempotent semirings the least fixed-point of a polynomial system of equations is equal to the least fixed-point of a linear system obtained by “linearizing” the polynomials of in a certain way. Our proofs rely on derivation tree analysis, a proof principle that combines methods from algebra, calculus, and formal language theory, and was first used in [5] to show that Newton’s method over commutative and idempotent semirings converges in a linear number of steps. Our results lead to efficient generic algorithms for computing the least fixed-point. We use these algorithms to derive several consequences, including an O (N 3) algorithm for computing the throughput of a context-free grammar (obtained by speeding up the O (N 4) algorithm of [2]), and a generalization of Courcelle’s result stating that the downward-closed image of a context-free language is regular [3].