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

Steven N. Karp

Wronskians, Total Positivity, and Real Schubert Calculus

Abstract. A complete flag in Rn is a sequence of nested subspaces V1 ⊂ ... ⊂ Vn-1 such that each Vk has dimension k. It is called totally nonnegative if all its Plücker coordinates are nonnegative. We may view each Vk as a subspace of polynomials in R[x] of degree at most n-1, by associating a vector (a1, ..., an) in Rn to the polynomial a1 + a2x + ... + anxn-1. We show that a complete flag is totally nonnegative if and only if each of its Wronskian polynomials Wr(Vk) 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(Vk) 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.

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