Todd's relation conjecture and binary relations for multiple zeta values in positive characteristic

@misc{hu2026todds,
  title = {{Todd's relation conjecture and binary relations for multiple zeta values in positive characteristic}},
  author = {Jinyuan Hu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.08263},
  eprint = {2609.08263},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08263},
}

TY  - UNPB
TI  - Todd's relation conjecture and binary relations for multiple zeta values in positive characteristic
AU  - Jinyuan Hu
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.08263
UR  - https://arxiv.org/abs/2609.08263
AB  - We prove Todd's relation conjecture: all 𝔽_q(θ)-linear relations of Thakur's multiple zeta values are generated from the fundamental binary relation by the operators ℬ^S, 𝒞^S, ℬ^S∘ 𝒞^S; moreover they are also generated by ℬ^∗S, 𝒞^S, ℬ^∗S∘𝒞^S. The ℬ^∗S-part of this conjecture has been proved by Chang, Chen and Mishiba. We prove the whole conjecture for Carlitz multiple polylogarithm values, which implies the ℬ^S-part for multiple zeta values. We also determine all fixed relations and binary relations. Let BR^S_w be the 𝔽_q(θ)-linear space spanned by binary relations of weight w, and let Fix^S_w be the 𝔽_q(θ)-linear space spanned by fixed relations. We derive generating functions ∑_w≥ 1(dim_𝔽_q(θ)Fix^S_w)x^w=(x^q+1(1−x))/((1−2x)(1−2x+x^q+1)) and ∑_w≥ 1(dim_𝔽_q(θ)BR^S_w)x^w=(x^q(1−x))/((1−2x)(1−2x+x^q+1)). Our results are based on the recent work of Im-Kim-Ngo Dac on the 𝔽_q-linear relations of Thakur's multiple zeta values, and a system of transfer theorems between multiple zeta values and multiple polylogarithm values.
ER  -