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Homological permanent vs. determinant

Prove that the permanent requires super‑polynomial algebraic circuits if and only if lim_{n→∞} h(perm_n)/h(det_n) = ∞, and moreover establish that perm ∉ VP if and only if h(perm_n) grows super‑polynomially in n.

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Background

Within their homological reformulation of algebraic complexity, the authors propose a criterion separating permanent from determinant based on comparative growth of homological complexity. This ties geometric complexity theory’s orbit‑closure viewpoint to homological invariants derived from computational complexes.

If validated, the conjecture would translate major algebraic lower‑bound questions (e.g., VP vs. VNP) into homological growth properties.

References

Conjecture [Homological Permanent vs. Determinant] The permanent function requires super-polynomial algebraic circuits if and only if: \lim_{n \to \infty} \frac{h(\mathrm{perm}_n)}{h(\mathrm{det}_n)} = \infty Moreover, \mathrm{perm} \notin VP is equivalent to h(\mathrm{perm}_n) growing super-polynomially in n. This conjecture unifies the geometric and topological approaches to the permanent vs. determinant problem:

— A Homological Proof of $\mathbf{P} \neq \mathbf{NP}$: Computational Topology via Categorical Framework (2510.17829 - Tang, 2 Oct 2025) in Conjecture (Homological Permanent vs. Determinant), Subsection "Connections with Geometric Complexity Theory" of Section 9 (Connections with Existing Theories)