Enumerative Real Algebraic Geometry

Two well-defined classes of structured polynomial systems have been studied from this point of view—sparse systems, where the structure is encoded by the monomials in the polynomials fi—and geometric systems, where the structure comes from geometry. This second class consists of polynomial formulations of enumerative geometric problems, and in this case Question 1.1 is the motivating question of enumerative real algebraic geometry, the subject of this survey. Treating both sparse polynomial systems and enumerative geometry together in the context of Question 1.1 gives useful insight. Given a system of polynomial equations (1.1) with d complex solutions, we know the following easy facts about its number ρ of real solutions.