{"id":"openai-math/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments","pdf":"https://github.com/openai/math/blob/main/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/prime-argument-polynomial-differences.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026","tags":["additive-combinatorics","analytic-number-theory","primes"],"score":0,"title":"A Power Saving for Polynomial Differences at Prime Arguments","domain":"Combinatorics","family":"182","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"Let h be a fixed integer polynomial of degree at least two with positive leading coefficient, having a unit root modulo every positive integer. We prove that any set $`A\\subseteq\\{1,\\ldots,N\\}`$ whose differences avoid all nonzero values $`h(p)`$ at primes satisfies $`|A|\\le C_hN^{1-c_h}`$, where $`c_h\\gt 0`$ and $`C_h\\ge1`$ depend only on h . Thus the local unit-root condition gives a fixed power saving even when polynomial arguments are restricted to primes. The proof uses the companion zero-free half-plane theorem for Dirichlet L -functions to obtain the required prime-distribution…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-05","reactions":{},"formalized":false,"family_title":"Power savings for intersective polynomial differences and prime arguments"}