Transcendental Hilbert-Kunz Multiplicities

Sudipta Das; Stephen Landsittel; Vinh Anh Pham

commutative algebra

Transcendental Hilbert-Kunz Multiplicities

arXiv:2607.28205

summary

The authors construct a normal graded domain over any uncountable algebraically closed field of characteristic p>2 together with a homogeneous primary ideal whose Hilbert‑Kunz multiplicity is a transcendental number, showing that such multiplicities can be non‑algebraic.

Abstract

We prove that ordinary Hilbert-Kunz multiplicity can be transcendental. More precisely, over every uncountable algebraically closed field of characteristic $p > 2$ , there exists a normal standard graded domain $S$ and an $S_{+}$ -primary homogeneous ideal $I \subseteq S$ such that $e_{HK} (I S_{S_{+}})$ is transcendental.

Topics & keywords

#hilbert-kunz multiplicity #transcendental numbers #graded rings #characteristic p #algebraic geometryHilbert-Kunz multiplicitytranscendentalnormal graded domaincharacteristic pprimary ideal