Transcendental Hilbert-Kunz Multiplicities

Sudipta Das; Stephen Landsittel; Vinh Anh Pham

Transcendental Hilbert-Kunz Multiplicities

Commutative Algebra 2026-07-30 v1 Algebraic Geometry

Authors: Sudipta Das , Stephen Landsittel , Vinh Anh Pham

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 $e_{\rm HK}(IS_{S_+})$ is transcendental.

Cite

@article{arxiv.2607.28205,
  title  = {Transcendental Hilbert-Kunz Multiplicities},
  author = {Sudipta Das and Stephen Landsittel and Vinh Anh Pham},
  journal= {arXiv preprint arXiv:2607.28205},
  year   = {2026}
}