Abstract: Many matroid invariants take values in polynomial rings. The study of the coefficients and roots of matroid invariants is an important topic in algebraic combinatorics. Examples are the characteristic polynomial, the Tutte polynomial or the Kazhdan-Lusztig polynomial of a matroid. Connections between matroid theory to the intersection theory of toric varieties has played an important role in the proof of several long-standing conjectures in recent years. In this talk I will explain techniques that have been successfully used to investigate properties of such polynomials.