Descartes' Rule of Signs for Polynomial Systems supported on Circuits
  We give a multivariate version of Descartes' rule of signs to bound the number of positive real roots of a system of polynomial equations in n variables with n+2 monomials, in terms of the sign variation of a sequence associated both to the exponent vectors and the given coefficients. We show that our bound is sharp and is related to the signature of the circuit. 
Fr\'ed\'eric Bihan and Alicia Dickenstein
math.AG cs.SC