Abstract
We prove the surjectivity part of Goncharov’s depth conjecture over a quadratically closed field. We also show that the depth conjecture implies that multiple polylogarithms of depth d and weight n can be expressed via a single function \({{\,\textrm{Li}\,}}_{n-d+1,1,\dots ,1}(a_1,a_2,\dots ,a_d)\), and we prove this latter statement for \(d=2\).
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References
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Acknowledgements
We are very grateful to Alexander Goncharov for suggesting Corollary 6 as a consequence of Theorem 5. SC was supported by Deutsche Forschungsgemeinschaft Eigene Stelle grant CH 2561/1-1, for Projektnummer 442093436, during the preparation of this work at Universität Hamburg. HG is grateful to the MPIM Bonn for excellent working conditions.
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Charlton, S., Gangl, H., Radchenko, D. et al. On the Goncharov depth conjecture and polylogarithms of depth two. Sel. Math. New Ser. 30, 27 (2024). https://doi.org/10.1007/s00029-024-00918-6
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DOI: https://doi.org/10.1007/s00029-024-00918-6