← 研踪 PaperTrailarXiv:2611.07280 · v1
Degree profiles of uniform polynomial consequences: Linear lower bounds for Nullstellensatz and Polynomial Calculus
首次提交 2026-11-05 09:01
版本更新 2026-11-05 09:01
雷达观测 2026-11-06 10:04
摘要
This paper proves asymptotically optimal linear degree lower bounds for Nullstellensatz and Polynomial Calculus, yielding a four-way polynomial-versus-exponential proof-size classification for uniform polynomial consequences over fields of positive characteristic. The size bounds use full, explicit coefficient lists, including zeros. Apart from a fixed finite initial range, exactly one of four cases occurs: $\mathrm T_1$, bounded degree; $\mathrm T_2$, degree at least linear in the universe size $n$; $\mathrm T_3$, constant degree on some residue classes and at least linear degree on the others; or $\mathrm T_4$, infinitely many recurring finite degrees, with bounded-degree subsequences and finite $Ω(n)$ peaks at arbitrarily large sizes. Thus the size patterns are polynomial bounds, exponential lower bounds, a periodic mixture, or polynomial subsequences with finite exponential peaks, respectively. Underivable targets have degree $\infty$, and both proof systems assign the same type. The result strengthens my 2008--2009 refutation classification and extends it to uniform consequences. The proof uses degree-preserving lifting with onset and period bounds linear in the degree. A common $p$-adic hardness spectrum describes the four types topologically. Distinguishing any two types is undecidable already for refutations. The linear degree bounds, four-way classification and pairwise undecidability results have been formalised and verified in Lean. The Lean formalisation, verification records and an accompanying tutorial are available in the supporting materials (https://doi.org/10.5281/zenodo.23965838).
作者备注：78 pages. Tutorial, Lean formalisation and verification records: https://doi.org/10.5281/zenodo.23965838
观测到的版本
相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。
v1 · 2026-11-05 09:01 · 最新观测版本
Degree profiles of uniform polynomial consequences: Linear lower bounds for Nullstellensatz and Polynomial Calculus
This paper proves asymptotically optimal linear degree lower bounds for Nullstellensatz and Polynomial Calculus, yielding a four-way polynomial-versus-exponential proof-size classification for uniform polynomial consequences over fields of positive characteristic. The size bounds use full, explicit coefficient lists, including zeros. Apart from a fixed finite initial range, exactly one of four cases occurs: $\mathrm T_1$, bounded degree; $\mathrm T_2$, degree at least linear in the universe size $n$; $\mathrm T_3$, constant degree on some residue classes and at least linear degree on the others; or $\mathrm T_4$, infinitely many recurring finite degrees, with bounded-degree subsequences and finite $Ω(n)$ peaks at arbitrarily large sizes. Thus the size patterns are polynomial bounds, exponential lower bounds, a periodic mixture, or polynomial subsequences with finite exponential peaks, respectively. Underivable targets have degree $\infty$, and both proof systems assign the same type. The result strengthens my 2008--2009 refutation classification and extends it to uniform consequences. The proof uses degree-preserving lifting with onset and period bounds linear in the degree. A common $p$-adic hardness spectrum describes the four types topologically. Distinguishing any two types is undecidable already for refutations. The linear degree bounds, four-way classification and pairwise undecidability results have been formalised and verified in Lean. The Lean formalisation, verification records and an accompanying tutorial are available in the supporting materials (https://doi.org/10.5281/zenodo.23965838).
查看此版本解读 →查看此版本原文 ↗