zbMATH — the first resource for mathematics

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:
all roots of each \(m_{k}\) in \(I\) are simple and belong to \(I_{1}\);
if \(k_{1}\neq k_{2}\), then \(m_{k_{1}}\) and \(m_{k_{2}}\) have no common roots;
for each \(k\), the function \(m_{k}\) admits precisely \(k(n-k)\) roots in \(I\).
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
Full Text: Link
[1] V. Arnold, On the number of flattening points on space curves, Sinai’s Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, vol. 171, Amer. Math. Soc., Providence, RI, 1996, pp. 11-22. MR 1359089 · Zbl 0973.53502
[2] I. Bárány, J. Matoušek, and A. Pór, Curves in R d intersecting every hyperplane at most d + 1 times, Computational geometry (SoCG’14), ACM, New York, 2014, pp. 565-571. MR 3382339 · Zbl 1395.05178
[3] A. Berenstein, S. Fomin, and A. Zelevinsky, Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122 (1996), no. 1, 49-149. MR 1405449 · Zbl 0966.17011
[4] G. Canuto, Associated curves and Plücker formulas in Grassmannians, Invent. Math. 53 (1979), no. 1, 77-90. MR 538685 · Zbl 0455.14019
[5] W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. MR 0460785
[6] V. Goulart and N. Saldanha, Combinatorialization of spaces of nondegenerate spherical curves, preprint arXiv:1810.08632 [math.GT].
[7] V. Goulart and N. C. Saldanha, Locally convex curves and the Bruhat stratification of the spin group, Isr. J. Math. (2021). DOI: 10.1007/s11856-021-2127-z · Zbl 07370908
[8] V. Goulart and N. C. Saldanha, Stratification by itineraries of spaces of locally convex curves, preprint arXiv:1907.01659 [math.GT].
[9] L. Lang, B. Shapiro, and E. Shustin, On the number of intersection points of the contour of an amoeba with a line, preprint arXiv:1905.08056 [math.AG]; to appear in Indiana Univ. Math. J. · Zbl 07403322
[10] A. J. Levin, The non-oscillation of solutions of the equation x (n) + p 1 (t)x (n−1) + • • • + pn(t)x = 0, Uspehi Mat. Nauk 24 (1969), no. 2 (146), 43-96 (Russian). MR 0254328. English translation: Russian Math. Surveys 24 (1969), no. 2, 43-99.
[11] V. Sedykh and B. Shapiro, On two conjectures concerning convex curves, Internat. J. Math. 16 (2005), no. 10, 1157-1173. MR 2182213 · Zbl 1085.14028
[12] B. Shapiro, Spaces of linear differential equations and flag manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 173-187, 223 (Russian). MR 1044054. English translation: Math. USSR-Izv. 36 (1991), no. 1, 183-197.
[13] B. Shapiro and M. Shapiro, Projective convexity in P 3 implies Grassmann convexity, Inter-nat. J. Math. 11 (2000), no. 4, 579-588. MR 1768174 · Zbl 1110.53301
[14] B. Shapiro and M. Shapiro, On the boundary of totally positive upper triangular matrices, Linear Algebra Appl. 231 (1995), 105-109. MR 1361102 · Zbl 0840.15012
[15] B. Shapiro and M. Shapiro, Linear ordinary differential equations and Schubert calculus, Proceedings of the Gökova Geometry-Topology Conference 2010, Int. Press, Somerville, MA, 2011, pp. 79-87. MR 2931881
[16] B. Shapiro, M. Shapiro, and A. Vainshtein, Connected components in the intersection of two open opposite Schubert cells in SLn(R)/B, Internat. Math. Res. Notices (1997), no. 10, 469-493. MR 1446839 · Zbl 0902.14035
[17] M. Shapiro, Nonoscillating differential equations, Ph.D. thesis, Moscow State University, 1992.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.