Degree profiles of uniform polynomial consequences: Linear lower bounds for Nullstellensatz and Polynomial Calculus

Abstract

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).