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Algebraic independence of the Carlitz period and the positive characteristic multizeta values at $ n$ and $ (n,n)$


Author: Yoshinori Mishiba
Journal: Proc. Amer. Math. Soc. 143 (2015), 3753-3763
MSC (2010): Primary 11J93; Secondary 11G09, 11M38
DOI: https://doi.org/10.1090/S0002-9939-2015-12532-4
Published electronically: February 25, 2015
MathSciNet review: 3359567
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Abstract: Let $ k$ be the rational function field over the finite field of $ q$ elements and $ \overline {k}$ its fixed algebraic closure. In this paper, we study algebraic relations over $ \overline {k}$ among the fundamental period $ \widetilde {\pi }$ of the Carlitz module and the positive characteristic multizeta values $ \zeta (n)$ and $ \zeta (n,n)$ for an ``odd'' integer $ n$, where we say that $ n$ is ``odd'' if $ q-1$ does not divide $ n$. We prove that these three elements are either algebraically independent over $ \overline {k}$ or satisfy some simple relation over $ k$. We also prove that if $ 2n$ is ``odd'', then these three elements are algebraically independent over $ \overline {k}$.


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Additional Information

Yoshinori Mishiba
Affiliation: Department of General Education, Oyama National College of Technology, 771 Nakakuki, Oyama, Tochigi, 323-0806, Japan
Email: mishiba@oyama-ct.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2015-12532-4
Received by editor(s): August 21, 2013
Received by editor(s) in revised form: April 24, 2014
Published electronically: February 25, 2015
Additional Notes: The author was partially supported by the JSPS Research Fellowships for Young Scientists
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society