2 1. OVERVIEW and complex numbers, respectively. For a positive integer n, write [n] for the set {1,...,n}, and let Zn be the free abelian group of rank n (a lattice), and Qn, Rn, and Cn, vector spaces of dimension n over the indicated fields. Likewise Pn and RPn are complex and projective spaces of dimension n, and Rn , (R∗)n, and (C∗)n = Tn for the groups of n-tuples of positive, nonzero real, and nonzero complex numbers, respectively. These groups, vector spaces, and projective spaces, all have distinguished ordered bases. We will use ZA, RA, TA, PA,... to denote the groups and spaces with distinguished bases indexed by the elements of a set A. 1.1. Introduction Our goal is to say something meaningful about the real solutions to a system of multivariate polynomial equations. For example, consider a system (1.1) f1(x1,...,xn) = f2(x1,...,xn) = · · · = fN(x1,...,xn) = 0 , of N real polynomials in n variables. Let r be its number of real solutions and let d be its number of complex solutions. We always assume that our systems are generic in the sense that all of their solutions are nondegenerate. Specifically, the differentials dfi of the polynomials at each solution span Cn, so that each solution has algebraic multiplicity 1. Our systems will come in families whose generic member is nondegenerate and has d complex solutions. Since every real number is complex, and since nonreal solutions come in complex conjugate pairs, we have the following trivial inequality, (1.2) d ≥ r ≥ d mod 2 ∈ {0, 1} . We can say nothing more unless the equations have some structure, and a partic- ularly fruitful class of structures are those which come from geometry. The main point of this book is that we can identify structures in equations that will allow us to do better than this trivial inequality (1.2). Our discussion will have three themes: (I) Sometimes, there is a smaller bound on r than d. (II) For many problems from enumerative geometry, the upper bound is sharp. (III) The lower bound on r may be significantly larger than d mod 2. A major theme will be the Shapiro Conjecture (Mukhin, Tarasov, and Varchenko Theorem [104]) and its generalizations, which is a situation where the upper bound of d is also the lower bound—all solutions to our system are real. This also occurs in Example 9.7. We will not describe how to actually find the solutions to a system (1.1) and there will be little discussion of algorithms and no complexity analysis. The book of Basu, Pollack, and Roy [4] is an excellent place to learn about algorithms for com- puting real algebraic varieties and finding real solutions. We remark that some tech- niques employed to study real solutions underlie numerical algorithms to compute the solutions [137]. Also, ideas from toric geometry [52, 61], Gr¨ obner bases [154], combinatorial commutative algebra [100], and Schubert Calculus [53] permeate this book. Other background material may be found in [31, 30].

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