Hilbert-Kunz series, F-signature series, and weak p-fractals

2025/10/22 by Caminata, Alessio, Zerman, Francesco
#Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.19355

Abstract

We extend the theory of p-fractals of Monsky and Teixeira by introducing the notion of weak p-fractal. We prove that for a hypersurface f having rational Hilbert-Kunz series is equivalent to the weak p-fractality of the associated function ϕ_f,p and having rational F-signature series is equivalent to the weak p-fractality of the reflection ϕ_f,p. In addition, we prove some results characterizing the shape of the generating series of numerical functions which are quasi-polynomials in pⁿ. This is motivated by the fact that the Hilbert-Kunz and F-signature functions take this form in several examples of interest.

Citations

• F-signature functions of diagonal hypersurfaces
• The Shape of Hilbert-Kunz Functions
• F-signature function of quotient singularities
• The Hilbert-Kunz functions of two-dimensional rings of type ADE
• Hilbert-Kunz functions over rings regular in codimension one
• F-Signature of Affine Toric Varieties
• Transcendence of some Hilbert-Kunz multiplicities (modulo a conjecture)
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• p-Fractals and power series--II. Some applications to Hilbert-Kunz theory
• Conic divisor classes over a normal monoid algebra
• The Hilbert-Kunz function in graded dimension two
• Hilbert-Kunz Functions for Normal Rings
• Semistability and Hilbert–Kunz multiplicities for curves
• Two theorems about maximal Cohen-Macaulay modules
• Simplicity of Rings of Differential Operators in Prime Characteristic
• Hilbert-Kunz functions of cubic curves and surfaces

Related

• p-Fractals and power series--II. Some applications to Hilbert-Kunz theory
• Syzygy gap fractals--I. Some structural results and an upper bound
• Limits of F-invariants and Riemann-Stieltjes integral
• F-signature functions of diagonal hypersurfaces
• A function determined by a hypersurface of positive characteristic