Wronskians, total positivity, and real Schubert calculus

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 , ∞ ) . In the language of Chebyshev systems, this means that the flag forms a Markov system or ECT -system on ( 0 , ∞ ) . 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 , ∞ ] . We use these results to show that a conjecture of Eremenko (Arnold Math J 1(3):339–342, 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 ( - ∞ , 0 ) , then all Plücker coordinates of V are real and positive. This conjecture is a totally positive strengthening of a result of Mukhin, Tarasov, and Varchenko. (J Am Math Soc 22(4):909–940, 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 version of the secant conjecture of Sottile (2003).