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# 1. Introduction

Consider the following question.

Question 1.1   Find a priori information about the number of real solutions to a structured system of real polynomial equations
 0 = f1 = f2 =  ...  = fN       for   fi   in   R[x1,x2,...,xn] (1.1)

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 r of real solutions.

 We have r < d,   and since d-r is an even integer, we have that r is at least 0    if d is even 1    if d is odd
(1.2)
In a surprising number of cases, much better information than this is known.

Structured systems occur in families of systems sharing the same structure. The common structure determines the number d of complex solutions to a general member of the family. We assume throughout that a general system in any family has only simple solutions in that each complex solution occurs without multiplicity.

Given such a family whose general member has d complex solutions, perhaps the ultimate answer to our motivating question is to determine exactly which numbers r of real solutions can occur and also which systems have a given number of real solutions. Because this level of knowledge may be unattainable, we will be satisfied with less knowledge.

For example, are the trivial bounds given in (1.2) sharp? That is, do there exist systems attaining the maximal and minimal number of real solutions allowed by (1.2)? If these bounds are not sharp, do there exist better sharp bounds? Perhaps we are unable to determine sharp bounds, but can exhibit systems in a family with many (or few) real solutions. This gives lower bounds on the maximum number of real solutions to a system in a family (or upper bounds on the minimum number).

These answers have two parts: bounds and constructions. We shall see that bounds (or other limitations) often come from topological considerations. On the other hand, the constructions often come by deformations from/to a degenerate situation. In enumerative geometry, this is the classical technique of special position, while for sparse systems, it is Viro's method of toric deformations.

In Section 2 we discuss sparse polynomial systems from the point of view of Question 1.1. The heart of this survey begins in Section 3, where we discuss some of the myriad examples of enumerative geometric problems that have been studied from this perspective. In particular, for many enumerative problems the upper bound (1.2) is sharp. In Section 4, we concentrate on enumerative problems from the Schubert calculus, where much work has been done on this question of real solutions. Section 5 is devoted to a conjecture of Shapiro and Shapiro, whose study has led to many recent results in this area. Finally, in Section 6 we describe new ideas of Eremenko and Gabrielov giving lower bounds better than (1.2) for some enumerative problems.

Next: 2. Polynomial Systems