The Special Schubert Calculus Is Real

We show that the Schubert calculus of enumerative geometry is real, for special Schubert conditions. That is, for any such enumerative problem, there exist real conditions for which all the a priori complex solutions are real. Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some general fixed figures [5]. For the problem of plane conics tangent to five general conics, the (surprising) answer is that all 3264 may be real [10]. Recently, Dietmaier has shown that all 40 positions of the Stewart platform in robotics may be real [2]. Similarly, given any problem of enumerating lines in projective space incident on some general fixed linear subspaces, there are real fixed subspaces such that each of the (finitely many) incident lines are real [13]. Other examples are shown in [12, 14], and the case of 462 4-planes meeting 12 general 3-planes in R is due to an heroic symbolic computation [4]. For any problem of enumerating p-planes having excess intersection with a collection of fixed planes, we show there is a choice of fixed planes osculating a rational normal curve at real points so that each of the resulting p-planes is real. This has implications for the problem of placing real poles in linear systems theory [1] and is a special case of a far-reaching conjecture of Shapiro and Shapiro [15]. Special Schubert conditions For background on the Grassmannian, Schubert cycles, and the Schubert calculus, see any of [8, 7, 6]. Let m, p ≥ 1 be integers. Let γ be a rational normal curve in R. For k > 0 and s ∈ γ, let Kk(s) be the k-plane osculating γ at s. For every integer a > 0, let τa(s) be the special Schubert cycle consisting of pplanes H in C which meet Km+1−a(s) nontrivially, and let τ(s) be the special Schubert cycle consisting of p-planes H in C meeting Km−1+a(s) improperly: dim H ∩Km−1+a(s) > a− 1. These cycles τa(s) and τ(s) each have codimension a and τ = τ1. Recall that the Grassmannian of p-planes in C has dimension Received by the editors December 20, 1998. 1991 Mathematics Subject Classification. Primary 14P99, 14N10, 14M15, 14Q20; Secondary 93B55.