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5.iii. Recent developments

Since this was written, we have found further evidence in support of these conjectures of Shapiro and Shapiro, and also more examples of enumerative problems are known that may have all their solutions real. In [So99], we show there is a choice of real numbers s₁, s₂, ..., s_mp in Conjecture 1 for which all d_m,p p-planes are real. More generally, the main result of [So99] is that for Pieri Schubert data in Conjecture 2, there is a choice of real numbers s₁, s₂, ..., s_n for which all p-planes are real.

We have also answered Question 1 concerning flag varieties affirmatively for hypersurface Schubert conditions. These are all Grassmanian Schubert data, where each condition comes from a hypersurface Schubert condition on a Grassmannian [So00b]. This leads to a conjectural anser to Question 2 for Grassmannian Schubert data involving non-crossing partitions (Manuscript and calcultions in preparation.) Similarly, a large class of enumerative problems arising in the quantum cohomology of flag manifolds (and related to systems theory) may have all their solutions be real [So00a]. The method of proof in these cases is related to the the homotopy continuation algorithms of [HSS]. In a related development, Dietmaier has shown that all 40 positions of the Stewart platform in robotics may be real [D].

A consequence of [So99] is that Conjecture 2 would follow from an affirmative answer to Question 4, namely that the systems which arise in these conjectures are always non-degenerate. While all this bolsters our conviction that these conjectures of Shapiro and Shapiro are true, the conjecture is still very much open. All of these results, and the evidence for these conjectures of Shapiro and Shapiro presented here, do show that there should be a broader theory of real enumerative geometry to explain these phenomena.

Very recently, A. Gabrielov and A. Eremenko gave a proof of Conjectures 1 and 2 for Grassmannians of 2-dimensioinal subspaces of a vector space [EG].