Hilbert–Kunz Multiplicity

• Hilbert–Kunz multiplicity is an asymptotic invariant defined for Noetherian local rings that quantifies the growth of Frobenius powers, serving as a numerical measure of singularity severity.
• It generalizes the Hilbert–Samuel multiplicity to modules and ideals through precise limits, with extensions including Amao-type and s-multiplicity theories under varied ring structures.
• Explicit formulas in cases like diagonal hypersurfaces and non-degenerate quadrics illustrate its use in computing singularities and guiding algebraic constructions in both geometric and arithmetic contexts.

Hilbert–Kunz multiplicity is an asymptotic invariant attached to Frobenius powers in prime characteristic. For a Noetherian local ring $(R,\mathfrak m)$ of characteristic $p>0$, dimension $d$, and an $\mathfrak m$-primary ideal $I$, the $q$-th Frobenius power is $I^{[q]}=(x^q\mid x\in I)$ for $q=p^e$, and Monsky showed that the limit

$$e_{\mathrm{HK}}(I)=\lim_{q\to\infty}\frac{\ell_R(R/I^{[q]})}{q^d}$$

exists. The resulting number measures the asymptotic growth of colengths under Frobenius and functions as a singularity invariant: $e_{\mathrm{HK}}(R)\ge 1$, and under mild hypotheses the equality $p>0$0 characterizes regularity (Landsittel et al., 29 Oct 2025, Datta et al., 2019).

1. Classical definition and structural properties

In its classical form, Hilbert–Kunz multiplicity is defined for $p>0$1-primary ideals, but the same asymptotic framework extends to finitely generated modules. If $p>0$2 is a finite $p>0$3-module and $p>0$4 is $p>0$5-primary, one writes

$p>0$6

with existence again attributed to Monsky (Smirnov, 2018). The invariant is closely related to Hilbert–Samuel multiplicity $p>0$7: the standard inequalities

$p>0$8

hold, and equality with $p>0$9 occurs for parameter ideals (Smirnov, 2018, Jeffries et al., 2020).

Several formal properties place Hilbert–Kunz multiplicity alongside the better-known multiplicity theories. It is additive in short exact sequences, and it satisfies an associativity formula over the top-dimensional associated primes: $d$0 In particular, for a domain $d$1 and a torsion-free module $d$2 of rank $d$3, one gets $d$4 (Jorge-Pérez et al., 2024). At the same time, Hilbert–Kunz multiplicity is subtler than Hilbert–Samuel multiplicity: it need not be an integer, and its numerical behavior is often more delicate in families (Smirnov, 2016).

The singularity-theoretic interpretation is central. The invariant is routinely described as measuring severity of singularities, values close to $d$5 indicating mild singularities, and in formally unmixed settings $d$6 characterizing regularity (Datta et al., 2019, Jeffries et al., 2020). This makes Hilbert–Kunz multiplicity a basic numerical interface between Frobenius asymptotics and local algebra.

2. Generalizations and interpolating invariants

A major development is the generalized Hilbert–Kunz theory for modules and for Frobenius-compatible families of ideals. Following Epstein–Yao, the generalized Hilbert–Kunz function of a finitely generated $d$7-module $d$8 is defined using the lengths of $d$9, and for ideals this can be described through saturation via

$\mathfrak m$0

The 2025 study of families of ideals introduces a $\mathfrak m$1-family $\mathfrak m$2 satisfying $\mathfrak m$3, and defines its generalized Hilbert–Kunz multiplicity by

$\mathfrak m$4

For the Frobenius family $\mathfrak m$5, this recovers the classical Hilbert–Kunz multiplicity (Landsittel et al., 29 Oct 2025).

That theory also introduces the Amao-type multiplicity

$\mathfrak m$6

for inclusions $\mathfrak m$7 with $\mathfrak m$8 of finite length. Under the uniform saturation condition $\mathfrak m$9, namely

$I$0

and suitable ring-theoretic hypotheses, the generalized Hilbert–Kunz multiplicity of a $I$1-family is realized as the asymptotic limit of Amao-type multiplicities. The proof is mediated by a volume-type theorem for inclusions of $I$2-families, using valuation-theoretic methods, local Okounkov bodies, and asymptotic lattice-point counting in cones (Landsittel et al., 29 Oct 2025).

Another line of development places Hilbert–Kunz multiplicity at one endpoint of a continuous interpolation. The $I$3-multiplicity compares ordinary powers and Frobenius powers through colengths of

$I$4

The associated limit exists, is continuous in $I$5, agrees with Hilbert–Samuel multiplicity in the small-$I$6 regime, and stabilizes at Hilbert–Kunz multiplicity for large $I$7. It also admits an associativity formula and an $I$8-closure theory interpolating between integral closure and tight closure (Taylor, 2017).

In mixed characteristic, a perfectoid analogue replaces Frobenius powers by perfectoidization and ordinary length by Faltings’ normalized length. For a complete Noetherian local domain $I$9 of mixed characteristic and an $q$0-primary ideal $q$1,

$q$2

defines the perfectoid Hilbert–Kunz multiplicity. In equal characteristic $q$3, this agrees with the classical invariant, and $q$4 if and only if $q$5 is regular (Cai et al., 2022).

3. Geometric variation and families

Hilbert–Kunz multiplicity is naturally viewed as a function on $q$6, via $q$7. For locally equidimensional $q$8-finite rings, and for locally equidimensional algebras essentially of finite type over an excellent local ring, this function is upper semi-continuous. The proof proceeds by establishing uniform convergence estimates for the normalized Hilbert–Kunz functions $q$9, so that semi-continuity for a fixed Frobenius level passes to the asymptotic limit (Smirnov, 2014).

A projective-geometric counterpart is a Bertini theorem. If $I^{[q]}=(x^q\mid x\in I)$0 is an equidimensional subscheme over an algebraically closed field of characteristic $I^{[q]}=(x^q\mid x\in I)$1, and $I^{[q]}=(x^q\mid x\in I)$2 for all $I^{[q]}=(x^q\mid x\in I)$3, then for a general hyperplane $I^{[q]}=(x^q\mid x\in I)$4 one has

$I^{[q]}=(x^q\mid x\in I)$5

This removes normality assumptions present in earlier work and shows that a uniform Hilbert–Kunz bound survives a general hyperplane cut (Datta et al., 2019).

The equimultiplicity theory for Hilbert–Kunz multiplicity replaces classical integral-closure criteria by tight-closure conditions. For an ideal $I^{[q]}=(x^q\mid x\in I)$6 and a parameter ideal $I^{[q]}=(x^q\mid x\in I)$7 modulo $I^{[q]}=(x^q\mid x\in I)$8, the formula

$I^{[q]}=(x^q\mid x\in I)$9

is equivalent to a colon-capturing condition asserting that for a system of parameters $q=p^e$0 modulo $q=p^e$1, each $q=p^e$2 is regular modulo $q=p^e$3 for all $q=p^e$4. Ideals satisfying these equivalent conditions are called Hilbert–Kunz equimultiple (Smirnov, 2016).

In the graded setting, the Hilbert–Kunz density function refines the multiplicity to a compactly supported continuous function $q=p^e$5 whose integral recovers the multiplicity: $q=p^e$6 This density is additive and satisfies a multiplicative formula for Segre products, making $q=p^e$7 accessible through Euclidean volume and piecewise-polynomial geometry in many projective situations (Trivedi, 2015).

A related characteristic-zero-oriented program studies the limit Hilbert–Kunz multiplicity

$q=p^e$8

For graded $q=p^e$9-primary ideals on smooth projective curves, the usual two-step limit can be replaced by the direct fixed-$$e_{\mathrm{HK}}(I)=\lim_{q\to\infty}\frac{\ell_R(R/I^{[q]})}{q^d}$$0 expression

$$e_{\mathrm{HK}}(I)=\lim_{q\to\infty}\frac{\ell_R(R/I^{[q]})}{q^d}$$1

which is obtained through syzygy bundles, cohomological estimates, and Harder–Narasimhan theory (Brenner et al., 2011).

4. Explicit formulas and asymptotic computations

A particularly rich source of exact formulas is the class of diagonal hypersurfaces

$$e_{\mathrm{HK}}(I)=\lim_{q\to\infty}\frac{\ell_R(R/I^{[q]})}{q^d}$$2

If $$e_{\mathrm{HK}}(I)=\lim_{q\to\infty}\frac{\ell_R(R/I^{[q]})}{q^d}$$3 with $$e_{\mathrm{HK}}(I)=\lim_{q\to\infty}\frac{\ell_R(R/I^{[q]})}{q^d}$$4, then

$$e_{\mathrm{HK}}(I)=\lim_{q\to\infty}\frac{\ell_R(R/I^{[q]})}{q^d}$$5

where $$e_{\mathrm{HK}}(I)=\lim_{q\to\infty}\frac{\ell_R(R/I^{[q]})}{q^d}$$6 is the Hilbert–Kunz multiplicity. For $$e_{\mathrm{HK}}(I)=\lim_{q\to\infty}\frac{\ell_R(R/I^{[q]})}{q^d}$$7, $$e_{\mathrm{HK}}(I)=\lim_{q\to\infty}\frac{\ell_R(R/I^{[q]})}{q^d}$$8 depends on $$e_{\mathrm{HK}}(I)=\lim_{q\to\infty}\frac{\ell_R(R/I^{[q]})}{q^d}$$9 in a subtle, nontrivial way; for $e_{\mathrm{HK}}(R)\ge 1$0, one has $e_{\mathrm{HK}}(R)\ge 1$1, and if some $e_{\mathrm{HK}}(R)\ge 1$2, then $e_{\mathrm{HK}}(R)\ge 1$3 (Gessel et al., 2010).

The same work reduces the asymptotic study of $e_{\mathrm{HK}}(R)\ge 1$4 as $e_{\mathrm{HK}}(R)\ge 1$5 to the first Frobenius level: $e_{\mathrm{HK}}(R)\ge 1$6 It then derives a closed limit formula in terms of explicit signed sums $e_{\mathrm{HK}}(R)\ge 1$7. In the quadratic case $e_{\mathrm{HK}}(R)\ge 1$8, the limit simplifies to the generating-function identity

$e_{\mathrm{HK}}(R)\ge 1$9

connecting Hilbert–Kunz multiplicity to Eulerian polynomials and classical analytic series (Gessel et al., 2010).

For non-degenerate quadrics

$p>0$00

the 2025 Ehrhart-theoretic analysis proves

$p>0$01

where $p>0$02 and $p>0$03 are Ehrhart polynomials of the Fibonacci and extended Fibonacci polytopes. Consequently, $p>0$04 is a rational function of $p>0$05, its limit as $p>0$06 is $p>0$07, and for fixed characteristic it is a decreasing function of the dimension $p>0$08 (Pak et al., 25 Aug 2025).

A complementary 2026 analysis of Fermat quadrics

$p>0$09

uses Green rings, tensor decompositions, and a Gelfand transform to derive analytic formulas and proves Yoshida’s conjecture that

$p>0$10

is decreasing on odd primes $p>0$11 (Meng, 3 Jun 2026).

5. Products, powers, and algebraic constructions

Hilbert–Kunz multiplicity behaves nontrivially under products of ideals. If $p>0$12 is quasi-unmixed, excellent, Noetherian, local, of characteristic $p>0$13, and $p>0$14 are $p>0$15-primary, then

$p>0$16

where $p>0$17 is the $p>0$18-spread of $p>0$19. When $p>0$20 has the same tight closure as a parameter ideal, equality holds if and only if $p>0$21. In the parameter-ideal case this yields the exact identity

$p>0$22

under the stated hypotheses (Epstein et al., 2015).

The asymptotic behavior of the powers $p>0$23 is governed by a second coefficient theory. For a local ring of characteristic $p>0$24, an $p>0$25-primary ideal $p>0$26, and a finitely generated module $p>0$27, the limit

$p>0$28

exists and controls the expansion

$p>0$29

This limit is additive on short exact sequences and satisfies a Northcott-type inequality in the Cohen–Macaulay case (Smirnov, 2018).

In dimension two, the Hilbert–Kunz multiplicity of powers is tightly linked to Ratliff–Rush closure. For a $p>0$30-dimensional excellent Cohen–Macaulay reduced local ring and an $p>0$31-primary ideal $p>0$32, the limit

$p>0$33

exists, and for all $p>0$34,

$p>0$35

Moreover, the eventual polynomial formula for $p>0$36 is equivalent to the eventual equality $p>0$37, which in turn is characterized by a tight-closure containment for Ratliff–Rush closures of Frobenius powers (Stefani et al., 2024).

Explicit ring constructions also admit clean formulas. For a fiber product $p>0$38, the Hilbert–Kunz multiplicity is determined by the dimensions and the multiplicities of $p>0$39, $p>0$40, and $p>0$41; in the top-dimensional case,

$p>0$42

For an idealization ring $p>0$43,

$p>0$44

These formulas produce structural consequences for regularity and lower bounds, and show that Hilbert–Kunz multiplicity is compatible with common singular ring constructions in a highly explicit way (Jorge-Pérez et al., 2024).

6. Extremal behavior, pathologies, and arithmetic features

Hilbert–Kunz multiplicity is neither universally rational nor uniformly tame in families. By interpreting Hilbert–Kunz theory for graded rings in terms of Frobenius asymptotics of cohomology on projective varieties, Brenner constructed three-dimensional quartic hypersurface domains and finite-length modules with irrational Hilbert–Kunz multiplicity, and deduced that the Hilbert–Kunz multiplicity of a local Noetherian domain can be irrational (Brenner, 2013).

The variation in families can also be wild. The equimultiplicity theory developed around the Brenner–Monsky hypersurface shows that Hilbert–Kunz multiplicity can attain infinitely many values, and that the equimultiple stratum need not be locally closed. In the specific hypersurface

$p>0$45

the set of values along primes above a fixed prime is infinite, and the stratum $p>0$46 is not locally closed (Smirnov, 2016).

These pathologies coexist with rigid rationality in combinatorial settings. For finitely generated semipositive cancellative reduced binoids, Hilbert–Kunz multiplicity exists, is rational, and is independent of the characteristic. Through the identification of binoid Hilbert–Kunz functions with the Hilbert–Kunz functions of the corresponding binoid algebras, this yields a characteristic-independent rationality theorem for a broad toric-combinatorial class (Batsukh et al., 2017).

The extremal problem of how close a singular ring can come to regularity organizes another large part of the theory. One conjectural picture, emphasized in the study of lower bounds, is that among non-regular formally unmixed local rings of fixed dimension, the smallest Hilbert–Kunz multiplicity should be attained by the $p>0$47 quadric singularity $p>0$48. In dimension $p>0$49, a sharp inequality is proved: $p>0$50 and equality forces strong $p>0$51-regularity and multiplicity $p>0$52; over an algebraically closed field with $p>0$53, equality identifies the determinantal quadric $p>0$54 (Jeffries et al., 2020). For hypersurfaces of multiplicity $p>0$55, especially the $p>0$56 singularities, one also has

$p>0$57

making the Hilbert–Kunz problem directly complementary to the extremal theory of $p>0$58-signature (Jeffries et al., 2020).