# Grassmann convexity and multiplicative Sturm theory, revisited

## TL;DR

This paper proves a special case of the Grassmann convexity conjecture, providing a formula for the maximum real zeros of Wronskians in disconjugate linear ODEs, and confirms its correctness for orders 4 and 5.

## Contribution

It establishes a conjectural formula for real zeros of Wronskians and proves its validity for certain differential equation orders, advancing understanding of Grassmann convexity.

## Key findings

- Proposed a formula for maximum real zeros of Wronskians.
- Confirmed the formula's correctness for equations of orders 4 and 5.
- Provided lower bounds for zeros in arbitrary order equations.

## Abstract

In this paper we settle a special case of the Grassmann convexity conjecture formulated earlier by B.and M.Shapiro. 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 $5$.