Deterministic polynomial factorization over prime fields

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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 \((n+1)\log p\). The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.

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What the papers state

Theorem 1.1

In Deterministic Polynomial Factorization over Prime Fields, 4 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

There is a uniform deterministic algorithm that, given a prime p in binary and a nonzero dense polynomial f\in\Fp[x] of degree n, outputs its leading coefficient and its distinct monic irreducible factors with their multiplicities in

O\bigl(((n+1)\lceil\log_2p\rceil)^{10^{12}}\bigr)

bit operations, with absolute implied constant. A constant polynomial returns an empty factor list. The algorithm uses no randomness, integer-factorization oracle, primitive-root oracle, or GRH assumption.

Theorem 1.2

In Deterministic Polynomial Factorization over Prime Fields, 4 October 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

There is a uniform deterministic algorithm with the following guarantee. Its input is a prime p and a nonzero dense polynomial f\in\Fp[x]. If p>B^{200000} and n\ge2, its input also includes one auxiliary prime \ell_q satisfying (1.2) for each prime q\le n. It outputs the complete factorization of f, including multiplicities. Let E=\max(2,\max_q\ell_q), with E=2 when no auxiliary primes are required. For an absolute constant C, the algorithm uses O((B+E)^C) bit operations. It uses neither randomness nor an integer factorization or primitive-root oracle.

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Papers

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Deterministic Polynomial Factorization over Prime Fields

4 October 2026, current edition.

Read it here PDF

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

Lean statements

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Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

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What these papers cite in the release, and what cites them

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Papers elsewhere in the release that this family’s papers cite

Primitive roots for every admissible integer baseCurrent edition; no accompanying Lean statement Family Primitive roots for every admissible integer baseCited by Deterministic Polynomial Factorization over Prime Fields.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

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28 references in Deterministic Polynomial Factorization over Prime Fields

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What people have said

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Identify the theorem the catalogue describes

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State the result in Lean

Family 142 The release has no Lean statement for this family. Open the form