2025/08/25 by Karp, Steven N.
#14M15 #14N15 #15B48 #26C10 #30C15 #34C10 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.18478
A classical result states that if f(z) is a polynomial of degree at most n with nonnegative coefficients, then f(z) has no zeros in the sector |arg(z)| < \fracπn of the complex plane, and the bound \fracπn is tight. Motivated by the Shapiro--Shapiro conjecture and related problems in real Schubert calculus, we generalize this result to Wronskians of polynomials. Namely, let f1(z), …, fk(z) be linearly independent polynomials of degree at most n whose coefficient matrix has all nonnegative k× k minors (that is, the polynomials span an element of the totally nonnegative Grassmannian in the sense of Lusztig and Postnikov). We show that the Wronskian polynomial Wr(f1, …, fk) has no complex zeros in the sector |arg(z)| < \fracπn (independent of k), and the bound \fracπn is tight. Our proof uses classical results of Gantmakher and Krein (1950) and Obreschkoff (1923) on sign variation.