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CS-111 · computer science · mathematics · Computational algebra and number theory · Computational complexity

Arithmetic-Formula Lower Bounds for Permanent and Determinant

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Prove superpolynomial lower bounds for the arithmetic-formula size of an explicit polynomial, in particular the permanent or determinant.

Formulation references: [S1], [S2].

Short English statement

Prove superpolynomial lower bounds for the arithmetic-formula size of an explicit polynomial, in particular the permanent or determinant.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-111-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-111

A cubic determinant-size lower bound gives polynomial-scale progress. The problem requires ruling out every polynomial size, and a fixed cubic exponent cannot establish that; broader permanent or determinant formula lower bounds also do not follow.

Arithmetic-Formula Lower Bounds for Permanent and Determinant

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 108; 2026-09-24), the asserted result is that complex exact and border determinant representations of the permanent require size of cubic order, with the stated branching-program consequences.

Mathematical statement of the claimed result

Manuscript 1.
Over $\mathbb C$, let $\operatorname{per}_m$ be the permanent of a generic $m\times m$ matrix. Its border determinantal complexity is the least $N$ for which it lies in the coefficientwise closure of determinants of $N\times N$ matrices with affine-linear entries. Exact determinantal complexity omits the closure. The asserted estimates are \[ \overline{\operatorname{dc}}(\operatorname{per}_m)\ge\frac{m^3}{5\,529\,600e},\qquad \operatorname{dc}(\operatorname{per}_m)\ge\frac{m^3}{5\,529\,600e}\quad(m\ge1408). \] The bound is cubic, rather than superpolynomial.

Outline of the argument

For manuscript 1 (family 108; 2026-09-24), smooth initial forms are tested by ambient polars. A characteristic-two adjugate subscheme determines a matrix on a fixed curve; normalization defects and finite-subscheme counts give matrix codimension bounds that persist under coefficientwise determinant limits.

Formal verification

For manuscript 1 (family 108; 2026-09-24), the documented Lean scope is cubic bounds for the permanent in both exact and border determinantal representations, together with the explicit large-size estimate and a smooth-initial-form criterion; the algebraic branching-program applications are outside the selected declarations.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are PermanentCubic.lean, SmoothInitialForm.lean.

Their named declarations include OAI.PermanentBorder.permanent_cubic_lower_bounds, OAI.PermanentBorder.smooth_initial_form_lower_bound.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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