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Grassmann convexity and multiplicative Sturm theory. Revisited. (English) Zbl 07406858
A linear ordinary homogeneous differential equation \[ y^{(n)}+p_{1}(x)y^{(n-1)}+\dots +p_{n}(x)y=0 \] of order \(n\) with real-valued continuous coefficients \(p_{i}(x)\) defined on interval \(I\subseteq\mathbb{R}\) is called disconjugate on \(I\) provided that any of its nontrivial solutions has at most \(n-1\) zeros on \(I\) counting multiplicities, where the interval \(I\) is open or closed. The second author of this paper proposed the following conjecture in the previous century [the second author, Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 1, 173–187 (1990; Zbl 0702.34032)].
Conjecture. (Upper bound on the number of real zeros of a Wronskian) Given any differential equation of the above form disconjugate on \(I\), a positive integer \(1\leq k\leq n-1\), and an arbitrary \(k\)-tuple \((y_{1}(x),y_{2}(x),\dots y_{k}(x))\) of its linearly independent solutions, the number of real zeros of \[ \det(W(y_{1}(x),y_{2}(x),\dots y_{k}(x)))=\det\left(\begin{matrix} y_{1}(x)& y_{1}^{\prime}(x)& \dots & y_{1}^{(k-1)}(x)\\ y_{2}(x)& y_{2}^{\prime}(x)& \dots & y_{2}^{(k-1)}(x)\\ \dots & \dots & \dots & \dots \\ y_{k}(x)& y_{k}^{\prime}(x)& & y_{k}^{(k-1)}(x)\end{matrix} \right) \] on \(I\) counting multiplicities does not exceed \(k(n-k)\).
The cases of \(k=1\) and \(k=n-1\) in the above conjecture are straightforward. The simplest non-trivial case that \(k=2\) and \(n=4\) has been settled in [the second and the third author, Int. J. Math. 11, No. 4, 579–588 (2000; Zbl 1110.53301)]. The first main result in this paper is
Theorem. The conjecture obtains for \(k=2\) and \(k=n-2\) for any \(n\geq3\).
The above conjecture has a reformulation. Let \(\mathrm{Lo}_{n}^{1}\) be the nilpotent Lie group of lower triangular \(n\times n\) matrices whose diagonal entries equal \(1\). Let \(\mathcal{J}\) be the set of \(n\times n\) real matrices \(A\) such that the entry \(A_{ij}\) is positive if \(i=j+1\) and \(0\) otherwise. A smooth curve \[ \Gamma:\left[ 0,1\right] \rightarrow\mathrm{Lo}_{n}^{1} \] is called flag-convex providing that \[ (\Gamma(t))^{-1}\Gamma^{\prime}(t)\in\mathcal{J} \] for all \(t\in\left[ 0,1\right] \). Given a flag-convex curve \(\Gamma\) and an integer \(k\) with \(0<k<n\), we define \(m_{k}:\left[ 0,1\right] \rightarrow \mathbb{R}\) by \[ m_{k}(t)=\det(\mathrm{swminor}(\Gamma(t),k)) \] for any \(t\in\left[ 0,1\right] \), where \(\mathrm{swminor}(L,k)\) is the \(k\times k\) submatrix of \(L\) formed by its last \(k\) rows and its first \(k\) columns. The above conjecture is now equivalent to
Conjecture. The number of zeros, counting multiplicies, of \(t\in\left[ 0,1\right] \) of the smooth function \(m_{k}:\left[ 0,1\right] \rightarrow\mathbb{R}\) is at most \(k(n-k)\).
The above theorem is of the following reformulation. Theorem. For any flag-convex function \(\Gamma:\left[ 0,1\right] \rightarrow \mathrm{Lo}_{n}^{1}\), the functions \(m_{2}\) and \(m_{n-2}\) have at most \(2(n-2)\) zeros each.
The second main result in this paper goes as follows.
Theorem. Consider a smooth flag-convex curve \(\Gamma_{\cdot}:I\rightarrow \mathrm{Lo}_{n}^{1}\) with a non-degenerate interval \(I\). Then, for any open subinterval \(I_{1}\subset I\), there exists a matrix \(L_{1}\in\mathrm{Lo} _{n}^{1}\) such that, for \(\Gamma_{1}(t)=L_{1}\Gamma_{\cdot }(t)\) and \(m_{k}=m_{\Gamma_{1},k}\), the following properties obtain:
(1)
all roots of each \(m_{k}\) in \(I\) are simple and belong to \(I_{1}\);
(2)
if \(k_{1}\neq k_{2}\), then \(m_{k_{1}}\) and \(m_{k_{2}}\) have no common roots;
(3)
for each \(k\), the function \(m_{k}\) admits precisely \(k(n-k)\) roots in \(I\).
MSC:
51M35 Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations)
34B05 Linear boundary value problems for ordinary differential equations
52A55 Spherical and hyperbolic convexity
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