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Mathematics of Computation

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On equiangular lines in $ 17$ dimensions and the characteristic polynomial of a Seidel matrix


Authors: Gary R. W. Greaves and Pavlo Yatsyna
Journal: Math. Comp. 88 (2019), 3041-3061
MSC (2010): Primary 05B20; Secondary 05B40, 05C45
DOI: https://doi.org/10.1090/mcom/3433
Published electronically: April 9, 2019
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Abstract: For $ e$ a positive integer, we find restrictions modulo $ 2^e$ on the coefficients of the characteristic polynomial $ \chi _S(x)$ of a Seidel matrix $ S$. We show that, for a Seidel matrix of order $ n$ even (resp., odd), there are at most $ 2^{\binom {e-2}{2}}$ (resp., $ 2^{\binom {e-2}{2}+1}$) possibilities for the congruence class of $ \chi _S(x)$ modulo $ 2^e\mathbb{Z}[x]$. As an application of these results we obtain an improvement to the upper bound for the number of equiangular lines in $ \mathbb{R}^{17}$, that is, we reduce the known upper bound from $ 50$ to $ 49$.


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Additional Information

Gary R. W. Greaves
Affiliation: School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
Email: grwgrvs@gmail.com

Pavlo Yatsyna
Affiliation: Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham, Surrey, TW20 0EX, United Kingdom
Email: pvyatsyna@gmail.com

DOI: https://doi.org/10.1090/mcom/3433
Received by editor(s): August 20, 2018
Received by editor(s) in revised form: January 14, 2019
Published electronically: April 9, 2019
Additional Notes: The first author was supported by the Singapore Ministry of Education Academic Research Fund (Tier 1); grant number: RG127/16.
Article copyright: © Copyright 2019 American Mathematical Society