Real Polynomial Systems
The development of the preceding three chapters focused on complex systems of homogeneous polynomial equations. The main algorithmic results in these chapters were satisfying: we can compute an approximate zero of a system f in average (and even smoothed) randomized polynomial time. Central in these results were the consideration of complex numbers for both the coefficients of the input system and the components of the computed approximate zero. For a variety of purposes, however, one is interested in real zeros of systems with real coefficients. Unlike the situation over the complex numbers, a system f of n real homogeneous polynomials in n+1 variables does not need to have a real solution. Therefore—and this is a situation we have already met when dealing with linear programming—the issue of feasibility precedes that of computing zeros.
In this chapter, we consider a problem more demanding than feasibility, namely, to count how many zeros the system has. Following our usual pattern, we say that a real system f is ill-posed when arbitrary small perturbations of f can change its number of real zeros. We then define the condition κ(f) as the relatived inverse distance of f to the set of ill-posed systems. The main goal of the chapter is to exhibit and analyze an algorithm for zero-counting in terms of κ(f).
We also present a short proof of a well-known result of Shub and Smale giving the expected number of real zeros for a random system f.
Finally, we briefly describe and analyze an algorithm for deciding feasibility of underdetermined systems of real polynomials.