Result 108, Theoretical computer science

A cubic permanent–determinant lower bound

Proves an Ω(n3)\Omega(n^3) lower bound for the border determinantal complexity of the n×nn\times n permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least cn3cn^3, for an absolute c > 0 and all sufficiently large n; the same bound therefore holds for exact representations.

Lean formalization New or sharp bound

The bigger picture

Why it matters

The permanent adds products of matrix entries much like a determinant, but without alternating signs. The manuscript reports that expressing it through determinants requires matrices whose size grows at least cubically, even when limits are allowed.

What changes?

For the n-by-n permanent over the complex numbers, the manuscript claims a lower bound of c times n cubed on the representing determinant's matrix size, for an absolute positive constant c and all sufficiently large n. Entries of that matrix may be arbitrary affine-linear expressions: constants plus linear combinations of the original entries. The bound also covers coefficientwise limits, where each polynomial coefficient converges to its target value. This limiting notion defines border determinantal complexity.

What does that help mathematicians do?

The claimed bound rules out smaller determinant representations even when exact equality is relaxed to convergence of coefficients. Exact representations are therefore ruled out as well. The abstract also reports cubic lower bounds on both vertices and edges in affine-linear algebraic branching programs, graphs that compute polynomials by summing products along paths. Those bounds include coefficientwise limits with a fixed vertex or edge budget, extending the obstruction to another structured model of algebraic computation.

Are there practical applications?

The immediate value is foundational: the result would sharpen limits on compressing the permanent into determinants or affine-linear branching programs. It distinguishes resources needed by these algebraic representations, including limiting ones. It does not establish a lower bound for every algorithm computing the permanent or demonstrate a practical speedup.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A cubic lower bound for border determinantal complexity of the permanent

September 24, 2026 35 pages

We prove that the complex border determinantal complexity of the m×mm\times m permanent is Ω(m3)\Omega(m^3), allowing arbitrary affine-linear determinant representations and coefficientwise limits. It also gives cubic lower bounds for exact determinantal complexity and for the numbers of vertices and edges in affine-linear algebraic branching programs, including coefficientwise limits with a fixed vertex or edge budget.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026,
  author = {{OpenAI}},
  title = {{A cubic lower bound for border determinantal complexity of the permanent}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026.pdf}{OAI:A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/108.md.

A cubic permanent–determinant lower bound

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves a cubic lower bound for both exact and border determinantal representations of the complex m×mm\times m permanent. For m≥1408m\ge1408, every affine-linear determinant representation of size nn, including coefficientwise limits, satisfies n≥m3/(5529600e)n\ge m^3/(5529600e).

A general supporting theorem applies to a polynomial in d≥2d\ge2 variables whose first nonzero homogeneous Taylor term has degree r≥2r\ge2 and no nonzero singular zero. Its exact and border determinantal sizes are at least (r−1)(d−1)/(4e)(r-1)(d-1)/(4e). The paper's algebraic-branching-program consequences are outside these statements.

Comparator links

Result Comparator statement
Cubic lower bounds for permanent determinantal complexity PermanentCubic.lean
Determinantal lower bound from a smooth initial form SmoothInitialForm.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.