Deterministic polynomial-time factorization of polynomials over prime fields

Lineage and corrections

Artin's primitive root entry, whose zero-free strip this proof uses evidence for this event

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Altman 2025, Deterministic polynomial factorisation modulo many primes (account of the problem) evidence for this event

Primitive roots for every admissible integer base (companion supplying the Hecke zero-free input) evidence for this event

Deterministic polynomial-time factorization of polynomials over prime fields evidence for this event

VibeMathed record: Deterministic polynomial-time factorization of polynomials over prime fields evidence for this event

Theorem 1.1: a uniform deterministic algorithm computes the complete factorization (irreducible factors with multiplicities) of any nonzero dense $f\in\mathbb F_p[x]$ in $O(((n+1)\lceil\log_2 p\rceil)^{10^{12}})$ bit operations, with no randomness, oracle or GRH. Theorem 1.2 (proved here, unconditional) does this given small auxiliary primes $\ell\equiv1\pmod{12q}$ with $p$ not a $q$th power mod $\ell$; the odd-degree splitter uses Jacobians of cyclic curves $Y^q=F(X)$ and norm equations in cyclic algebras. Small auxiliary primes are deduced from a uniform zero-free strip $\mathrm{Re}\,s>1-10^{-6}$ for all finite-order Hecke L-functions over cyclotomic fields containing $\mu_{12}$, imported from the companion on primitive roots and not proved here. The exponent is not practical. parent of this event

Deterministic polynomial-time factorization of polynomials over prime fields parent of this event