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Math-1501 · mathematics · Graph theory

O16 — Factoring polynomials modulo a prime

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Statement

Definitions and mathematical statement

The input is a prime $p$ and a nonzero polynomial $f\in\mathbb F_p[X]$, given by its dense coefficient list. A complete irreducible factorization means an equality \[ f=c\prod_{i=1}^t f_i^{e_i}, \] where $c\in\mathbb F_p^\times$, the $f_i$ are pairwise distinct monic irreducible polynomials of positive degree, and the $e_i$ are positive integers. Irreducible means that $f_i$ has no factorization into two positive-degree polynomials over $\mathbb F_p$. Constants have an empty factor list.

Is there a deterministic classical algorithm computing such a factorization for every input in time polynomial in $d=\deg f$ and $\log p$? All field elements are represented by their standard binary residues and factors are output densely. Thus the cost counts bit operations rather than treating arbitrarily large field operations as free.

The task includes multiplicities, including inseparable behavior when the characteristic divides relevant exponents. The order in which factors are listed is immaterial. The prime is allowed to vary, and the coefficients need not come from reductions of one previously fixed integer polynomial.

The sought guarantee is unconditional and worst-case. Algorithms that are randomized, work only for fixed characteristic, depend on an unproved hypothesis, or achieve a polynomial cost averaged across many primes do not establish this uniform deterministic bound for each individual pair $(p,f)$.

Short English statement

Can every polynomial over every prime field be factored completely by an unconditional deterministic polynomial-time algorithm?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1501-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1501

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 142; 2026-10-04).

O16 — Factoring polynomials modulo a prime

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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