  We develop several notions of multiplicity for linear factors of
multivariable polynomials over different arithmetics (hyperfields). The key
example is multiplicities over the hyperfield of signs, which encapsulates the
arithmetic of $\mathbf{R}/\mathbf{R}_{>0}$. These multiplicities give us
various upper and lower bounds on the number of linear factors with a given
sign pattern in terms of the signs of the coefficients of the factored
polynomial. Using resultants, we can transform a square system of polynomials
into a single polynomial whose multiplicities give us bounds on the number of
positive solutions to the system. In particular, we are able to re-derive the
lower bound of Itenberg and Roy on any potential upper bound for the number of
solutions to a system of equations with a given sign pattern. In addition, our
techniques also explain a particular counterexample of Li and Wang to Itenberg
and Roy's proposed upper bound.
