The threshold for linear independence of multiple zeta values in positive characteristic
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911628222529536 |
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| author | Im, Bo-Hae Kim, Hojin Dac, Tuan Ngo |
| author_facet | Im, Bo-Hae Kim, Hojin Dac, Tuan Ngo |
| contents | 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 $\mathbb{F}_q$. 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 $\mathbb{F}_q$-linear relation. This result provides the first counterexample to Thakur's conjecture. Our proof relies on a new connection between MZVs and Carlitz multiple polylogarithms over $\mathbb{F}_q$, generalizing a central result of [IKLNDP24]. We also introduce a modification of the algorithm from [ND21] that yields a weight-preserving operator acting on $\mathbb{F}_q$-linear relations, providing the algebraic framework for these results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25451 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The threshold for linear independence of multiple zeta values in positive characteristic Im, Bo-Hae Kim, Hojin Dac, Tuan Ngo Number Theory Primary 11M32, Secondary 11G09, 11J93, 11M38, 11R58 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 $\mathbb{F}_q$. 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 $\mathbb{F}_q$-linear relation. This result provides the first counterexample to Thakur's conjecture. Our proof relies on a new connection between MZVs and Carlitz multiple polylogarithms over $\mathbb{F}_q$, generalizing a central result of [IKLNDP24]. We also introduce a modification of the algorithm from [ND21] that yields a weight-preserving operator acting on $\mathbb{F}_q$-linear relations, providing the algebraic framework for these results. |
| title | The threshold for linear independence of multiple zeta values in positive characteristic |
| topic | Number Theory Primary 11M32, Secondary 11G09, 11J93, 11M38, 11R58 |
| url | https://arxiv.org/abs/2604.25451 |