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OP-6125Status unassessedComputational complexity

Lower Bounds for Rational Formulas Computing the Permanent

TheoremDB P3564 · complexity theory

MSC 68Q · Theory of computing

Inferred from the community field; finer classification has not been reviewed.

Source catalogue assessmentAs of 2026-09-08Source & evidence ↓
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Problem statement

For n≥32n\ge32, prove that every valid rational arithmetic formula over C\mathbb C computing the n×nn\times n permanent has at least n4/(192log⁡2n)n^4/(192\log_2 n) variable leaves.

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Formal statement

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theorem PermanentFormulaLowerBound.permanent_rational_formula_logarithmic_lower_bound {n : ℕ} (hn : 32 ≤ n) (f : RationalFormula (Fin n × Fin n) ℂ) (hvalid : RationalFormula.Valid f) (hf : RationalFormula.eval f = algebraMap (MvPolynomial (Fin n × Fin n) ℂ) (FractionRing (MvPolynomial (Fin n × Fin n) ℂ)) (permanentPolynomial n)) : (n : ℝ) ^ 4 / (192 * Real.logb 2 (n : ℝ)) ≤ (RationalFormula.variableLeaves f : ℝ)

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Progress & source assessment

Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.

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References
  1. [1]TheoremDB P3564 link
  2. [2]Source formal statement (not locally verified) link

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Discussion

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