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PreprintCC BY 4.0arXiv

Multiplying matrices via polynomials: exposition and barriers

  • Alimzhan Amanov
  • Maxim van den Berg
  • Jeroen Zuiddam
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Abstract

We give an exposition of OpenAI's recent proof that the exponent ωC of matrix multiplication over the complex numbers satisfies ωC≤9/4. Our version of the proof gives ωF≤9/4 over every field 𝔽. We also treat the rectangular matrix multiplication exponent ωF(a,b,c). Here we have ωF(1,k,1)=2 for 0≤ k≤1/2 and ωF(1,k,1)≤1+k+1/(4k) for k≥1/2, and, for real numbers a≥ b≥ c≥0, ωF(a,b,c)=a+b if c(a+b)≤ ab and ωF(a,b,c)≤(a+b+c)/2+(ab/c+ac/b+bc/a)/4 otherwise. Our exposition is in the language of degeneration inequalities among polynomial multiplication tensors and Strassen's asymptotic spectrum duality. We moreover prove an operational barrier (which can be thought of as coming from an operational asymptotic spectrum) showing that the current analysis of all these upper bounds on ωF and ωF(a,b,c) is optimal and that in fact the current degeneration ingredients cannot improve the bounds.

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Cite

BibTeXRIS
BibTeX
@misc{amanov2026multiplying,
  title = {{Multiplying matrices via polynomials: exposition and barriers}},
  author = {Alimzhan Amanov and Maxim van den Berg and Jeroen Zuiddam},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30054},
  eprint = {2610.30054},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30054},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Multiplying matrices via polynomials: exposition and barriers
AU  - Alimzhan Amanov
AU  - Maxim van den Berg
AU  - Jeroen Zuiddam
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.30054
UR  - https://arxiv.org/abs/2610.30054
AB  - We give an exposition of OpenAI's recent proof that the exponent ω_C of matrix multiplication over the complex numbers satisfies ω_C≤9/4. Our version of the proof gives ω_F≤9/4 over every field 𝔽. We also treat the rectangular matrix multiplication exponent ω_F(a,b,c). Here we have ω_F(1,k,1)=2 for 0≤ k≤1/2 and ω_F(1,k,1)≤1+k+1/(4k) for k≥1/2, and, for real numbers a≥ b≥ c≥0, ω_F(a,b,c)=a+b if c(a+b)≤ ab and ω_F(a,b,c)≤(a+b+c)/2+(ab/c+ac/b+bc/a)/4 otherwise. Our exposition is in the language of degeneration inequalities among polynomial multiplication tensors and Strassen's asymptotic spectrum duality. We moreover prove an operational barrier (which can be thought of as coming from an operational asymptotic spectrum) showing that the current analysis of all these upper bounds on ω_F and ω_F(a,b,c) is optimal and that in fact the current degeneration ingredients cannot improve the bounds.
ER  -