Descartes' rule of signs for arbitrary fewnomial systems

Source abstract

We consider systems of $n$ real polynomial equations in $n$ variables with $n+k+1$ monomials. By the Gale duality of Bihan and Sottile, their positive solutions correspond to the solutions of a system of $k$ equations $\prod_ip_i^{B_{ij}}=1$ in a polyhedron $Δ\subset\mathbb{R}^k$, where the $p_i$ are affine functions, and a Khovanskii--Rolle argument bounds their number by the number of common zeros in $Δ$ of iterated Jacobians $Γ_k,\dots,Γ_1$ plus the number of noncompact branches of certain curves. We bound the first term by the Bézout number minus the numbers of zeros in the other chambers of the arrangement $\{p_i=0\}$, which we bound from below by boundary degrees given by a facet-count formula. The branches of the curves end at zeros of the Jacobians on faces of $Δ$, which we count on the flats of the arrangement through explicit reduced systems. The resulting recursion over all flats and chambers starts on lines with Descartes' rule of signs for circuits. We obtain upper bounds for the number of positive solutions which only depend on the oriented matroid of the coefficient matrix and on the oriented matroids of the exponent matrix and of its liftings.