  We study a family of symmetric polynomials that we refer to as the Boolean
product polynomials. The motivation for studying these polynomials stems from
the computation of the characteristic polynomial of the real matroid spanned by
the nonzero vectors in $\mathbb{R}^n$ all of whose coordinates are either $0$
or $1$. To this end, one approach is to compute the zeros of the Boolean
product polynomials over finite fields. The zero loci of these polynomials cut
out hyperplane arrangements known as resonance arrangements, which show up in
the context of double Hurwitz polynomials. By relating the Boolean product
polynomials to certain total Chern classes of vector bundles, we establish
their Schur-positivity by appealing to a result of Pragacz relying on earlier
work on numerical positivity by Fulton-Lazarsfeld. Subsequently, we study a
two-alphabet version of these polynomials from the viewpoint of
Schur-positivity. As a special case of these polynomials, we recover symmetric
functions first studied by D\'esarm\'enien and Wachs in the context of descents
in derangements.
