Polynomial hitting lists for noncommutative rational formulas
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For noncommutative rational formulas over Q with unrestricted inverse nesting, can a deterministic polynomial-time algorithm produce a polynomial-size list of matrix evaluations hitting every nonzero formula of bounded size?
Why it matters and what counts as progress
Why it matters: Polynomial lists would improve the quasipolynomial black-box bounds known for this rational-formula model.
What counts as progress: A list with polynomial dimension and total bit length on which each admissible nonzero formula has at least one defined invertible evaluation.
How the problem developed
- 4 May 2022 · Partial result [1]
Vikraman Arvind, Abhranil Chatterjee and Partha Mukhopadhyay Constructed deterministic quasipolynomial-time hitting sets for rational formulas of inversion height two, the first nested-inverse case.
- 2024 · Partial result [2]
Vikraman Arvind, Abhranil Chatterjee and Partha Mukhopadhyay Reported quasipolynomial-size black-box hitting sets for rational formulas of arbitrary inversion height.
- 24 Sept 2026 · Proof claim [3]
OpenAI Released a manuscript claiming deterministic polynomial-time construction of polynomial-size rational-matrix hitting lists for admissible nonzero noncommutative rational formulas.
AI contribution
Polynomial Hitting Lists for Noncommutative Rational Formulas
Given unary n and a size bound s, a deterministic machine constructs in polynomial bit time a polynomial-size list of rational matrix tuples of polynomial total output length and common polynomial dimension. Every admissible nonzero noncommutative rational formula over Q of size at most s has a defined, invertible evaluation on a listed tuple.
Sources
- Arvind, Chatterjee and Mukhopadhyay, Black-Box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial Time
- Arvind, Chatterjee and Mukhopadhyay, Black-Box Identity Testing of Noncommutative Rational Formulas in Deterministic Quasipolynomial Time
- OpenAI, Polynomial Hitting Lists for Rational Formulas
- Polynomial Hitting Lists for Noncommutative Rational Formulas
- Rational hitting-list formalization scope
- Lean rational hitting-list theorem
Website timeline
Publication dates and editorial changes.
OpenAI reports polynomial hitting lists for rational formulas