# Polynomial value density reference

Family 020: Squarefree quartics and power-free polynomial values. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Arithmetic sampling researchers can compare squarefree or power-free outputs of admissible irreducible polynomials to predicted Euler-product densities.

## Applicability and commercial boundary

The local-divisor exclusion and degree ranges are essential. Positive limiting density supplies no finite sample rate or fast guaranteed generator.

## Initial business decision

Research infrastructure or conditional engineering until a computable implementation, effective bounds and a recurring buyer need are demonstrated. No commercial novelty or profitability is established.

## Next verification action

Extract effective error terms and a polynomial with computable local factors.

## Evidence scope

The catalog statement was individually reviewed. Manuscript proof and selected formal scope comparison remain queued. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Squarefree values of quartics and power-free values of polynomials](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/manuscript.pdf).