Finiteness of rank for Grassmann convexity

TL;DR

This paper investigates the Grassmann convexity conjecture, providing a general explicit upper bound for the maximal number of real zeros of Wronskians in solutions to disconjugate linear ODEs, linking it to convex curves in a nilpotent group.

Contribution

The paper establishes a general explicit upper bound for the Grassmann convexity conjecture, advancing understanding of zeros of Wronskians in linear differential equations.

Findings

Confirmed the conjecture's bounds in small dimensions

Derived a new explicit upper bound for the conjecture

Linked convex curves in nilpotent groups to differential equations

Abstract

The Grassmann convexity conjecture gives 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. The conjecture can be reformulated in terms of convex curves in the nilpotent lower triangular group. The formula has already been shown to be a correct lower bound and to give a correct upper bound in several small dimensional cases. In this paper we obtain a general explicit upper bound.