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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
Citations:
Zbl 0702.34032; Zbl 1110.53301
Full Text:
References:
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