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arXiv · 2607.10763

The rank of the modular multiplication matrix

Haojia Shi
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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.

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Haojia Shi. 2026-07-12. The rank of the modular multiplication matrix. https://arxiv.org/abs/2607.10763

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