We prove that if all roots of the discrete Wronskian with step 1 of a set of quasi-exponentials with real bases are real, simple and differ by at least 1, then the complex span of this set of quasi-exponentials has a basis consisting of quasi-exponentials with real coefficients. This theorem generalizes the statement of the B. and M. Shapiro conjecture about spaces of polynomials.
The proof is based on the Bethe ansatz method for the XXX model.
@article{CML_2009__1_2_225_0,
author = {Evgenii Mukhin and Vitaly Tarasov and Aleksandr Varchenko},
title = {On reality property of {Wronski} maps},
journal = {Confluentes Mathematici},
pages = {225--247},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {1},
number = {2},
year = {2009},
doi = {10.1142/S1793744209000092},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744209000092/}
}
TY - JOUR AU - Evgenii Mukhin AU - Vitaly Tarasov AU - Aleksandr Varchenko TI - On reality property of Wronski maps JO - Confluentes Mathematici PY - 2009 SP - 225 EP - 247 VL - 1 IS - 2 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744209000092/ DO - 10.1142/S1793744209000092 LA - en ID - CML_2009__1_2_225_0 ER -
%0 Journal Article %A Evgenii Mukhin %A Vitaly Tarasov %A Aleksandr Varchenko %T On reality property of Wronski maps %J Confluentes Mathematici %D 2009 %P 225-247 %V 1 %N 2 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744209000092/ %R 10.1142/S1793744209000092 %G en %F CML_2009__1_2_225_0
Evgenii Mukhin; Vitaly Tarasov; Aleksandr Varchenko. On reality property of Wronski maps. Confluentes Mathematici, Tome 1 (2009) no. 2, pp. 225-247. doi : 10.1142/S1793744209000092. https://cml.centre-mersenne.org/articles/10.1142/S1793744209000092/
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