2024/05/24 by Steven N. Karp, E. Mukhin, Karp, Steven N. +3 · 1 citation
Mathematics · Medicine · #05E05 #14M15 #15B48 #30C15 #82B23 #Advanced Algebra and Geometry #Combinatorics (math.CO) #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Medical Imaging Techniques and Applications #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2405.20229
openalex publication_date 2024/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A quasi-exponential is an entire function of the form ecup(u), where p(u) is a polynomial and c ∈ ℂ. Let V = ⟨ eh1up1(u), …, ehNupN(u) ⟩ be a vector space with a basis of quasi-exponentials. We show that if h1, …, hN are nonnegative and all of the complex zeros of the Wronskian Wr(V) are real, then V is totally nonnegative in the sense that all of its Grassmann-Plücker coordinates defined by the Taylor expansion about u=t are nonnegative, for any real t greater than all of the zeros of Wr(V). Our proof proceeds by showing that the higher Gaudin Hamiltonians TλG(t) introduced in [ALTZ14] are universal Plücker coordinates about u=t for the Wronski map on spaces of quasi-exponentials. The result that V is totally nonnegative follows from the fact that TλG(t) is positive semidefinite, which we establish using partial traces. We also show that if h1 = ⋯ = hN = 0 then TλG(t) equals βλ(t), which is the universal Plücker coordinate for the Wronski map on spaces of polynomials introduced in [KP23].