Moscow Mathematical Journal

Volume 21, Issue 3, July–September 2021 pp. 613–637.

Grassmann Convexity and Multiplicative Sturm Theory, Revisited

Authors: Nicolau Saldanha (1), Boris Shapiro (2), and Michael Shapiro (3)
Author institution:(1) Departamento de Matemática, PUC-Rio R. Mq. de S. Vicente 225, Rio de Janeiro, RJ 22451-900, Brazil
(2) Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
(3) Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA

Summary:

In this paper we settle a special case of the Grassmann convexity conjecture formulated by the second and the third authors about a decade ago. We present a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. We show that this formula gives the lower bound for the required total number of real zeros for equations of an arbitrary order and, using our results on the Grassmann convexity, we prove that the aforementioned formula is correct for equations of orders 4 and 5.

2020 Math. Subj. Class. Primary: 34B05; Secondary: 52A55.

Keywords: Disconjugate linear ordinary differential equations, Grassmann curves, osculating flags, Schubert calculus.

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