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CS-110 · computer science · mathematics · Computational algebra and number theory

Three zeta-like products, degree-count inversion, and the factorization boundary

Status Partially solved · unsolvedmath.comLLM status Open unverified

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.
Can one compute the zeta function of $\operatorname{Spec}(\mathbb Z/n\mathbb Z)$, \[\zeta_n(s)=\prod_{p\mid n}(1-p^{-s})^{-1},\] without first factoring $n$? For a polynomial $f\in\mathbb F_q[x]$, consider the analogous product over its monic irreducible factors. From this zeta data, can one recover the factorization of $f$ in deterministic polynomial time?

Formulation references: [S1], [S2].

Short English statement

Can one compute the zeta function of $\operatorname{Spec}(\mathbb Z/n\mathbb Z)$, \[\zeta_n(s)=\prod_{p\mid n}(1-p^{-s})^{-1},\] without first factoring $n$? For a polynomial $f\in\mathbb F_q[x]$, consider the analogous product over its monic irreducible factors. From this zeta data, can one recover the factorization of $f$ in deterministic polynomial time?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-110-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-110

Dense prime-field polynomial factorization addresses the polynomial branch directly. The integer zeta-product and integer-factorization task, together with the arbitrary-extension-field formulation, remains outside the proved algorithmic scope.

Three zeta-like products, degree-count inversion, and the factorization boundary

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 142; 2026-10-04), the asserted result is that dense polynomials over binary-specified prime fields have uniform deterministic complete factorization in bit time polynomial in degree and $\log p$, using the cited Hecke zero-free input.

Mathematical statement of the claimed result

Manuscript 1.
For any binary-encoded prime $p$ and nonzero densely encoded $f\in\mathbb F_p[x]$ of degree $n$, a uniform deterministic algorithm is claimed to output \[ f=c\prod_i g_i^{e_i} \] with $c$ the leading coefficient, distinct monic irreducible $g_i$ and positive integer multiplicities, in $O(((n+1)\lceil\log_2p\rceil)^{10^{12}})$ bit operations. Constants have an empty factor list. The computation is unconditional and deterministic; it receives neither an integer-factorization service nor a primitive-root oracle.

The separate algebraic reduction sets $L=\lceil\log_2p\rceil$, $B=20+(n+1)(L+1)$. If $p>B^{200000}$ and $n\ge2$, supply for every prime $q\le n$ a prime $\ell_q\notin\{2,3,p,q\}$ with \[ \ell_q\equiv1\pmod{12q},\qquad p^{(\ell_q-1)/q}\not\equiv1\pmod{\ell_q}. \] That reduction has bit cost $O((B+E)^C)$ for absolute $C$, where $E=\max(2,\max_q\ell_q)$, and $E=2$ when no such primes are required. This bound depends on the numerical size of the auxiliary primes; the full theorem additionally constructs sufficiently small primes.

Outline of the argument

For manuscript 1 (family 142; 2026-10-04), primary generators and norm equations reduce splitting to an odd-degree separator. Lifted ramification classes are filtered to keep extension degrees polynomial, and divisors at infinity distinguish roots; the analytic input bounds the auxiliary primes required by the finite algebraic algorithm.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 142; 2026-10-04).

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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