Factoring multivariate polynomials over hyperfields and the multivariable Descartes' problem

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 $\mathbf{R}\mathrm{/}{\mathbf{R}}_{>0}$. 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.