Reality property of discrete Wronski map with imaginary step
type: Item
instance of: scholarly article
title: Reality property of discrete Wronski map with imaginary step
zbMATH Open document ID: 1299.81027
author: Vitaly O. Tarasov
author: E. Mukhin
author: A. Varchenko
published in: Letters in Mathematical Physics
publication date: 2012-09-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1102.5368
review text: This work generalizes a conjecture of B. and M. Shapiro. It follows from the Fundamental Theorem of Algebra that a polynomial \(p(x)\in \mathbb{C}[x]\) has real coefficients (up to an overall constant) if all its roots are real. The B. and M. Shapiro conjecture that if the Wronskian of \(N\) polynomials with complex coefficients has real roots only, then the space spanned by these polynomials has a basis consisting of polynomials with real coefficients. This conjecture was proved in [\textit{E. Mukhin} et al., Ann. Math. (2) 170, No. 2, 863--881 (2009; Zbl 1213.14101)]. The authors consider the discrete Wronskian with purely imaginary step \(2h\) of a space of \(N\) quasi-exponentials \(e^{\lambda_ix}p_i(x)\) with \(\lambda_i\in\mathbb{R}\) et \(p_i\in \mathbb{C}[x]\). They show that if this discrete Wronskian is real and for every its roots the imaginary part is at most \(|h|\), then the the space of quasi-exponentials has a basis such that all polynomials have real coefficients. The proof is based on the Bethe ansatz for the \(XXX\) model and is similar to the method of [Zbl 1213.14101] and [\textit{E. Mukhin} et al., Confluentes Math. 1, No. 2, 225--247 (2009; Zbl 1181.81065)]. The conjecture of B. and M. Shapiro is reobtained taking the limit \(h\rightarrow 0\). Several other generalizations of this conjecture are known, see [\textit{A. Eremenko} et al., Proc. Am. Math. Soc. 134, No. 4, 949--957 (2006; Zbl 1110.14052)] and [Zbl 1181.81065].
reviewed by: Alessandro Ruzzi
Mathematics Subject Classification ID: 81R12
Mathematics Subject Classification ID: 14P05
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 82B23
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 13J30
zbMATH DE Number: 6084517
zbMATH Keywords: discrete Wronski map
zbMATH Keywords: B. and M. Shapiro conjecture
zbMATH Keywords: Bethe ansatz
zbMATH Keywords: \(XXX\) model
MaRDI profile type: Publication
OpenAlex ID: W3099541032
arXiv ID: 1102.5368
cites work: Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry
cites work: Rational functions and real Schubert calculus
cites work: Q3361250
cites work: The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
cites work: On reality property of Wronski maps
cites work: Generating operator of \(XXX\) or Gaudin transfer matrices has quasi-exponential kernel
cites work: Representations of Yangians with GelfandZetlin bases
DOI: 10.1007/S11005-011-0521-X
description: scientific article; zbMATH DE number 6084517