The main purpose of this paper is to introduce a new category, which we call a resonance category, whose combinatorics reflect that of canonical stratifications of n -fold symmetric smash products. The study of the stratifications can then be abstracted to the study of functors satisfying certain sets of axioms, which we name resonance functors. One frequently studied stratification is that of the set of all polynomials of degree n , defined by fixing the allowed multiplicities of roots. We apply our abstract combinatorial framework, in particular, the notion of direct product of relative resonances, to study the Arnold problem of computing the algebro-topological invariants of these strata.