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PreprintCC BY-NC-ND 4.0HAL

The threshold for linear independence of multiple zeta values in positive characteristic

  • Bo-Hae Im
  • Hojin Kim
  • Tuan Ngo Dac
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Abstract

A fundamental conjecture formulated by Thakur in 2009, which has guided significant developments in function field arithmetic, asserts that multiple zeta values (MZV's) in positive characteristic of fixed weight are linearly independent over F_q. In this paper we settle this conjecture by determining the precise threshold for this independence. We prove that linear independence holds for all weights up to 2q, while for weight 2q + 1 we establish the existence of a unique and explicit F_q-linear relation. This result provides the first counterexample to Thakur's conjecture. Our proof relies on a new connection between MZV's and Carlitz multiple polylogarithms over F_q, generalizing a central result of [31]. We also introduce a modification of the algorithm from [38] that yields a weight-preserving operator acting on F_q-linear relations, providing the algebraic framework for these results.

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Cite

BibTeXRIS
BibTeX
@misc{im2026threshold,
  title = {{The threshold for linear independence of multiple zeta values in positive characteristic}},
  author = {Bo-Hae Im and Hojin Kim and Tuan Ngo Dac},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  eprint = {2604.25451},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05604679v2},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - The threshold for linear independence of multiple zeta values in positive characteristic
AU  - Bo-Hae Im
AU  - Hojin Kim
AU  - Tuan Ngo Dac
PY  - 2026
DA  - 2026/09/10
PB  - HAL
UR  - https://hal.science/hal-05604679v2
AB  - A fundamental conjecture formulated by Thakur in 2009, which has guided significant developments in function field arithmetic, asserts that multiple zeta values (MZV's) in positive characteristic of fixed weight are linearly independent over F_q. In this paper we settle this conjecture by determining the precise threshold for this independence. We prove that linear independence holds for all weights up to 2q, while for weight 2q + 1 we establish the existence of a unique and explicit F_q-linear relation. This result provides the first counterexample to Thakur's conjecture. Our proof relies on a new connection between MZV's and Carlitz multiple polylogarithms over F_q, generalizing a central result of [31]. We also introduce a modification of the algorithm from [38] that yields a weight-preserving operator acting on F_q-linear relations, providing the algebraic framework for these results.
ER  -