Result 142, Theoretical computer science

Deterministic polynomial factorization over prime fields

Gives a uniform deterministic algorithm that completely factors every nonzero dense degree-n polynomial over a prime field đ”œp, including multiplicities, in bit complexity polynomial in (n+1)log⁥p(n+1)\log p. The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.

Algorithm or complexity result

The bigger picture

Why it matters

Factoring a polynomial means breaking it into smaller pieces that cannot be split further. The manuscript claims this can always be done efficiently over arithmetic modulo a prime, without relying on random choices.

What changes?

The manuscript reports a single deterministic algorithm for every prime p supplied in binary and every nonzero degree-n polynomial supplied as its full coefficient list. Arithmetic is modulo p, the prime field setting. It returns all irreducible factors, which cannot be split further over that field, and their multiplicities. Its bit-operation count is polynomial in (n+1) log p. It requires no randomness, integer-factorization oracle, primitive-root oracle, or Generalized Riemann Hypothesis assumption.

What does that help mathematicians do?

The claimed bound would make complete factorization a deterministic polynomial-time task measured against the dense input's size, even when the prime is large. Researchers could use it as a subroutine without introducing randomness or leaving repeated factors unresolved. The proof depends on the uniform Hecke zero-free theorem claimed in the companion paper, "Primitive roots for every admissible integer base", so that theorem is essential to the reported guarantee.

Are there practical applications?

Its immediate value is foundational: it would establish a worst-case deterministic efficiency guarantee for a basic task in exact algebra. It could support symbolic computations that need polynomial factors over prime fields. The supplied abstract gives no concrete runtime estimates or implementation evidence, so the polynomial bound alone does not establish practical speed.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Deterministic Polynomial Factorization over Prime Fields

October 4, 2026 48 pages

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∈Fp[x]f\in\mathbf 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 (deg⁥f+1)log⁥p(\deg f+1)\log p. The proof uses the uniform Hecke zero-free theorem from the companion paper *Primitive roots for every admissible integer base*.

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Cite (BibTeX) 
@misc{OAI:Deterministic-Polynomial-Factorization-over-Prime-Fields-October-4-2026,
  author = {{OpenAI}},
  title = {{Deterministic Polynomial Factorization over Prime Fields}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Deterministic-Polynomial-Factorization-over-Prime-Fields-October-4-2026/Deterministic-Polynomial-Factorization-over-Prime-Fields.pdf}{OAI:Deterministic-Polynomial-Factorization-over-Prime-Fields-October-4-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.