OpenAI · 142-01 · 2026-10-04 · Theoretical computer science

Deterministic Polynomial Factorization over Prime Fields

preprints/Deterministic-Polynomial-Factorization-over-Prime-Fields-October-4-2026

Family 142. Deterministic polynomial factorization over prime fields.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Gives a uniform deterministic algorithm that completely factors every nonzero dense degree- $n$ polynomial over a prime field $F_{p}$ , including multiplicities, in bit complexity polynomial in $(n + 1) lo g p$ . The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.

We give a uniform deterministic polynomial-time algorithm for complete factorization over prime fields. For a prime p in binary and a nonzero polynomial $f \in F_{p} [x]$ given by its dense coefficient list, the algorithm computes the irreducible factors and their multiplicities using a number of bit operations polynomial in $(de g f + 1) lo g p$ . The proof uses the uniform Hecke zero-free theorem from the companion paper *Primitive roots for every admissible integer base*.

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.