2023/05/11 by Jason McCullough, McCullough, Jason · 1 citation
Mathematics · #13D02 #13D05 #13P20 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2305.06532
openalex publication_date 2023/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Ananyan and Hochster proved the existence of a function Φ(m,d) such that any graded ideal I generated by m forms of degree at most d in a standard graded polynomial ring satisfies reg(I) ≤ Φ(m,d). Relatedly, Caviglia et. al. proved the existence of a function Ψ(e) such that any nondegenerate prime ideal P of degree e in a standard graded polynomial ring over an algebraically closed field satisfies reg(P) ≤ Ψ(deg(P)). We provide a construction showing that both Φ(3,d) and Ψ(e) must be at least doubly exponential in d and e, respectively. Previously known lower bounds were merely super-polynomial in both cases.