we-solve (beta)

CS-108 · computer science · mathematics · Computational algebra and number theory

Finite-field factorization and sparse-polynomial root testing

Status Open · 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.
For a degree-$n$ polynomial over $\mathbb F_q$, is factorization possible deterministically in time polynomial in $n$ and $\log q$? Determine the unconditional complexity of factoring $x^2-a$, and the optimal exponent in randomized factorization over $\mathbb F_2$. For a collection of polynomials of degree two or three, is deciding whether all split into linear factors faster through their product or through separate tests? Finally, can one decide whether $x^\alpha+ax^\beta+b$ has a root in $\mathbb F_q$ in time polynomial in $\log q$, where $0<\beta<\alpha\leq q-1$ and $a,b\in\mathbb F_q$?

Formulation references: [S1], [S2].

Short English statement

For a degree-$n$ polynomial over $\mathbb F_q$, is factorization possible deterministically in time polynomial in $n$ and $\log q$? Determine the unconditional complexity of factoring $x^2-a$, and the optimal exponent in randomized factorization over $\mathbb F_2$. For a collection of polynomials of degree two or three, is deciding whether all split into linear factors faster through their product or through separate tests? Finally, can one decide whether $x^\alpha+ax^\beta+b$ has a root in $\mathbb F_q$ in time polynomial in $\log q$, where $0<\beta<\alpha\leq q-1$ and $a,b\in\mathbb F_q$?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-108-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-108

Complete dense factorization over prime fields is covered, as are its quadratic applications. The bundle additionally asks about arbitrary finite fields, optimal randomized exponents, batched comparisons, and sparse-exponent root tests; a prime-field factorization theorem does not resolve every one of those tasks.

Finite-field factorization and sparse-polynomial root testing

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

Go to contribution →