Ananyan and Hochster proved the existence of a function $\Phi (m,d)$ such that any graded ideal $I$ generated by $m$ forms of degree at most $d$ in a standard graded polynomial ring satisfies $\mathrm{reg}\left(I\right)\le \Phi (m,d)$. Relatedly, Caviglia et. al. proved the existence of a function $\Psi \left(e\right)$ such that any nondegenerate prime ideal $P$ of degree $e$ in a standard graded polynomial ring over an algebraically closed field satisfies $\mathrm{reg}\left(P\right)\le \Psi (deg(P\left)\right)$. We provide a construction showing that both $\Phi (3,d)$ and $\Psi \left(e\right)$ must be at least doubly exponential in $d$ and $e$, respectively. Previously known lower bounds were merely super-polynomial in both cases.

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Jason McCullough ^{1}

@article{CRMATH_2024__362_G3_251_0, author = {Jason McCullough}, title = {Prime {Ideals} and {Three-generated} {Ideals} with {Large} {Regularity}}, journal = {Comptes Rendus. Math\'ematique}, pages = {251--255}, publisher = {Acad\'emie des sciences, Paris}, volume = {362}, year = {2024}, doi = {10.5802/crmath.544}, language = {en}, }

Jason McCullough. Prime Ideals and Three-generated Ideals with Large Regularity. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 251-255. doi : 10.5802/crmath.544. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.544/

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