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Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro. (English) Zbl 0997.14016

From the introduction: The determination of the number of real solutions to a system of polynomial equations is a challenging problem in symbolic and numeric computation with real world applications. Related questions include when a problem of enumerative geometry can have all solutions real and when may a given physical system be controlled by real output feedback. In May 1995, Boris Shapiro and Michael Shapiro communicated to the author a remarkable conjecture connecting these three lines of inquiry. Their conjecture is false – we give full description and present a counterexample in section 5. However, there is considerable evidence for their conjecture if the Schubert cells are in a Grassmann manifold. Here is the simplest (but still very interesting and open) special case of this conjecture: Let \(m,p>1\) be integers and let \(X\) be a \(p\times m\)-matrix of indeterminates. Let \(K(s)\) be the \(m \times (m+p)\)-matrix of polynomials in \(s\) whose \((i,j)\)-th entry is \({j-i \choose i-1} s^{j-i}\). Set \(\varphi_{m,p} (s;X): =\det {K(s) \brack I_p\;X}\), where \(I_p\) is the \(p\times p\) identity matrix.
Conjecture 1.1 (Shapiro and Shapiro). For all integers \(m,p>1\), the polynomial system \(\varphi_{m,p} (1;X)= \varphi_{m,p} (2;X) =\cdots= \varphi_{m,p} (mp;X)=0\) is zero-dimensional with \[ d_{m,p}: ={1!2!3! \cdots(p-2)! (p-1)!\cdot (mp)!\over m!(m+1)!(m+2)! \cdots (m+p-1)!} \] solutions, and all of them are real.
A. Eremenko and A. Gabrielov give a proof of this conjecture when either \(m\) or \(p\) is 2 [Ann. Math. (2) 155, No. 1, 105–129 (2002; see the preceding review Zbl 0997.14014)]. – Conjecture 1.1 is related to a question of Fulton, who asked how many solutions to a problem of enumerative geometry may be real, where that problem consists of counting figures of some kind having a given position with respect to some given (fixed) figures. More examples, including that of 3-planes in \(\mathbb{C}^6\) meeting 9 given 3-planes nontrivially, are found by F. Sottile [in: Algebraic geometry, Proc. Summer Res. Inst., Santa Cruz 1995, Proc. Symp. Pure Math. 62 (pt. 1), 435–447 (1997; Zbl 0890.14030) and J. Pure Appl. Algebra 117/118, 601–615 (1997; Zbl 0889.14026)].

MSC:

14N15 Classical problems, Schubert calculus
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14Q15 Computational aspects of higher-dimensional varieties
14P99 Real algebraic and real-analytic geometry
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