The rank of the modular multiplication matrix

Haojia Shi

The rank of the modular multiplication matrix

Number Theory 2026-07-12 v1

Authors: Haojia Shi

Abstract

We prove Bueno et al.'s conjectural formula for the rank of the modular multiplication matrix for every positive integer. The proof first applies the additive Fourier transform to split the row space by exact additive character order, and then applies the multiplicative character decomposition on unit groups to count the nonzero components. The same formula holds after deleting the zero row and zero column.

Keywords

modular multiplication matrix matrix rank additive Fourier transform multiplicative character number theory

Cite

@article{arxiv.2607.10763,
  title  = {The rank of the modular multiplication matrix},
  author = {Haojia Shi},
  journal= {arXiv preprint arXiv:2607.10763},
  year   = {2026}
}

Comments

7 pages, no figures