Prime Ideals and Three-generated Ideals with Large Regularity - McCullough, Jason (Comptes Rendus. Mathématique)
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 $reg(I) \le Φ (m,d)$. 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 $reg(P) \le Ψ (\deg (P))$. 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.