Products of arithmetic matroids and quasipolynomial invariants of CW-complexes

In this note we prove that the product of two arithmetic multiplicity functions on a matroid is again an arithmetic multiplicity function. This allows us to answer a question by Bajo–Burdick–Chmutov [2], concerning the modified Tutte–Krushkal–Renhardy polynomials defined by these authors. Furthermore, we show that the Tutte quasi-polynomial introduced by Brändén and Moci encompasses invariants defined by Beck–Breuer–Godkin–Martin [3] and Duval–Klivans–Martin [11] and can thus be considered as a dichromate for CW complexes.