Reality property of discrete Wronski map with imaginary step
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].
scientific article; zbMATH DE number 6084517
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