#####
Séminaire Lotharingien de Combinatoire, 86B.20 (2022), 10 pp.

# Steven N. Karp

# Wronskians, Total Positivity, and Real Schubert Calculus

**Abstract.**
A complete flag in **R**^{n} is a sequence of nested subspaces
*V*_{1} ⊂ ... ⊂ *V*_{n-1} such that each
*V*_{k} has dimension *k*. It is called *totally nonnegative* if all its Plücker coordinates are nonnegative. We may view each *V*_{k} as a subspace of polynomials in **R**[*x*] of degree at most *n*-1, by associating a vector (*a*_{1}, ..., *a*_{n}) in **R**^{n} to the polynomial *a*_{1} + *a*_{2}*x* + ... + *a*_{n}*x*^{n-1}. We show that a complete flag is totally nonnegative if and only if each of its Wronskian polynomials Wr(*V*_{k}) is nonzero on the interval (0, infinity). In the language of Chebyshev systems, this means that the flag forms a Markov system or *ECT*-system on (0, infinity). This gives a new characterization and membership test for the totally nonnegative flag variety. Similarly, we show that a complete flag is totally positive if and only if each Wr(*V*_{k}) is nonzero on [0, infinity]. We use these results to show that a conjecture of Eremenko (2015) in real Schubert calculus is equivalent to the following conjecture: if *V* is a finite-dimensional subspace of polynomials such that all complex zeros of Wr(*V*) lie in the interval (-infinity, 0), then all Pl\"{u}cker coordinates of *V* are real and positive. This conjecture is a totally positive strengthening of a result of Mukhin, Tarasov, and Varchenko (2009), and can be reformulated as saying that all complex solutions to a certain family of Schubert problems in the Grassmannian are real and totally positive. We also show that our conjecture is equivalent to a totally positive strengthening of the secant conjecture (2012).

Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.

The following versions are available: