Wronskians, total positivity, and real Schubert calculus

• Steven N. Karp ^[1]
  1. [1] University of Notre Dame

     University of Notre Dame

     Township of Portage, Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 1, 2024, págs. 1-28
• Idioma: inglés
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• Resumen

  • 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 +···+ an xn−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,∞). 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(Vk ) 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).