@inbook{e6a2994f83d348389f1fcae1c80f5632,
title = "The Shape of Hilbert–Kunz Functions",
abstract = "We discuss Hilbert–Kunz function from when it was originally defined to its recent developments. A brief history of Hilbert–Kunz theory is first recounted. Then we review several techniques involved in the study of Hilbert–Kunz functions by presenting some illustrative proofs without going into details of the technicalities. The second part of this article focuses on the Hilbert–Kunz function of an affine normal semigroup ring and relates it to Ehrhart quasipolynomials. We pay extra attention to its periodic behavior and discuss how the cellular decomposition constructed by Bruns and Gubeladze fits into the computation of the functions. The closed forms of the Hilbert–Kunz function of some examples are presented. The discussion in this part highlights the close relationship between Hilbert–Kunz function and Ehrhart theory.",
keywords = "Affine semigroup ring, Characteristic p, Cohen–Macaulay cones, Conic divisor, Divisor class group, Ehrhart{\textquoteright}s theorem, Frobenius homomorphism, Harder-Narasimhan filtration, Hilbert–Kunz function, Hilbert–Kunz multiplicity, Local Riemann–Roch formula, Quasipolynomial, Representation ring",
author = "Chan, {C. Y.Jean}",
note = "Funding Information: My sincere gratitude goes to Kazuhiko Kurano for showing me the connection between Hilbert–Kunz functions and Ehrhart Theory. To Joseph Gubeladze for pointing to the beautiful source [6]. To Uli Walther for discussions on the Macaulay 2 codes. To Holger Brenner and Claudia Miller for insightful conversations offering support and many crucial references. Special thanks to I-Chiau Huang and Academia Sinica of Taiwan who have sponsored multiple visits throughout the course while the main ideas in this article were formed. I am grateful for the anonymous referees for providing detailed comments and corrections that greatly improved the manuscript. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.",
year = "2021",
doi = "10.1007/978-3-030-91986-3_5",
language = "English",
series = "Association for Women in Mathematics Series",
publisher = "Springer Science and Business Media Deutschland GmbH",
pages = "111--163",
booktitle = "Association for Women in Mathematics Series",
}