---
title: F-pure Threshold (FPT) Overview
url: https://www.emergentmind.com/topics/f-pure-threshold-fpt
type: topic
---

# F-pure Threshold (FPT) Overview

The F-pure threshold (FPT) is a fundamental invariant in the study of singularities in prime characteristic commutative algebra and algebraic geometry. It measures how tightly a given ideal or hypersurface fails to split under Frobenius endomorphism, providing deep connections between arithmetic, singularity theory, and birational geometry. The F-pure threshold is the characteristic $p$ analogue of the log-canonical threshold (LCT) in characteristic zero, and its precise computation illuminates subtle arithmetic and geometric properties of singularities.

## 1. Rigorous Definition and Fundamental Properties

Let $R$ be an $F$-finite Noetherian ring of characteristic $p > 0$ and let $I \subset R$ be an ideal containing a nonzerodivisor. For a real number $t > 0$, the pair $(R, I^t)$ is sharply $F$-split if, for infinitely many $e \gg 0$, the natural map
\[
I^{\, t(p^e - 1)} \cdot \mathrm{Hom}_R(F_*^e R, R) \longrightarrow R
\]
is surjective. The **F-pure threshold** is defined as
\[
\mathrm{fpt}(R, I) = \sup\{ t > 0 \mid (R, I^t)\text{ is sharply $F$-split} \}.
\]
In the case where $R = k[x_1, \ldots, x_n]$ is a regular local or polynomial ring, this can be equivalently described as
\[
\mathrm{fpt}(I) = \lim_{e \to \infty} \frac{\nu_I(p^e)}{p^e}, \quad \nu_I(p^e) = \max\{ r \mid I^r \not\subseteq \mathfrak{m}^{[p^e]} \},
\]
where $\mathfrak{m}^{[p^e]} = (x_1^{p^e}, \dots, x_n^{p^e})$ [2506.18891, 1210.6729].

Key properties include:
- **Rationality:** F-pure thresholds are rational numbers.
- **Semicontinuity:** The FPT behaves upper semi-continuously in families [2501.13613].
- **Comparison with LCT:** If $f \in \mathbb{Q}[x_1,\ldots,x_n]$, then for all large $p$,
  \[
  \mathrm{fpt}(f_p) \leq \mathrm{lct}(f), \quad \lim_{p \to \infty}\mathrm{fpt}(f_p) = \mathrm{lct}(f).
  \]
- **Regularity Criterion:** $\mathrm{fpt}(R) = \dim(R)$ if and only if $R$ is regular [2501.13613].
- **Discreteness and ACC:** The set of F-pure thresholds is discrete and satisfies the ascending chain condition (ACC) [1404.3772, 2501.13613].

## 2. Relationship to Characteristic Zero Invariants

The F-pure threshold is closely related to the log-canonical threshold, a birational invariant of singularities in characteristic zero. For smooth $n$-dimensional varieties, Bhatt–Hernández–Miller–Mustaţă establish that the set of all limit points of sequences of F-pure thresholds as $p \to \infty$ coincides with the set of LCTs of ideals in dimension $n$ [1106.0207]:
\[
 \lim_{p \to \infty} \mathrm{fpt}_p(a_p) = \mathrm{lct}_0(a_0).
\]
This provides a dictionary between $F$-singularities in positive characteristic and log-canonical thresholds in characteristic zero, mediated via the reduction mod $p$ process. The F-pure threshold also controls the behavior of test ideals, which are characteristic $p$ analogues of multiplier ideals in characteristic zero.

## 3. Explicit Computations and Structural Formulas

### a) Monomials and Binomials

For monomial ideals, $\mathrm{fpt}((x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n})) = \min_{i} (1/a_i)$ [1106.0207]. For binomial hypersurfaces, explicit formulas for the FPT are given in terms of the splitting polytope defined by the exponent vectors and their base-$p$ expansions, with the value determined by a "no-carry" criterion on the base-$p$ digits and the position of critical points in the polytope [1112.2427].

### b) Homogeneous Polynomials in Two Variables: Truncation Formulas

For homogeneous degree $d$ polynomials in two variables, significant progress has been made in precisely enumerating possible F-pure thresholds. Smith and Vraciu establish that the maximum FPT among all degree $d$ forms is given by a truncation of the base $p$ expansion of $2/d$, and that all truncations occurring at places determined by the order of $p$ modulo $d$ can arise as thresholds of suitable reduced forms:
\[
\mathrm{fpt}_{\mathrm{gen}}(2, d, p) =
  \begin{cases}
    1, & d \leq 2, \\
    A_L, & d > 2,
  \end{cases}
\]
where $A = 2/d$ and $A_L$ is the $L$-th truncation of its (non-terminating) base-$p$ expansion. Existence theorems for reduced forms realize every truncation at a suitable place as an FPT, refining earlier necessary conditions [2207.11800].

A counterexample demonstrates that the necessary list of congruence conditions from Hernández–Núñez-Betancourt–Witt–Zhang is not sufficient for the realization of a given truncation as an FPT, and Smith–Vraciu conjecture that the true constraint is the order of $p$ in the multiplicative group $(\mathbb{Z}/d)^{\times}$ [2207.11800].

### c) Generic and Extremal Bounds

For homogeneous polynomials of degree $d$ in $n$ variables, the generic FPT is the largest truncation $A_L$ satisfying two arithmetic properties depending on $n,d,p$; in most cases, the generic value is strictly below the characteristic zero log canonical threshold $n/d$ [2207.11800].

A general sharp lower bound for reduced forms of degree $d$ is given by the first nonzero digit in the base-$p$ expansion of $2/d$ in two variables or $n/d$ generally:
\[
\mathrm{fpt}(f) \geq \text{first nonzero truncation of }\frac{n}{d}
\]
with sharpness established in a wide range of degrees and characteristics [2207.11800].

### d) Structure of Denominators

For homogeneous two-variable forms, it is shown that if $\mathrm{fpt}(f) \ne 2/d$, then the denominator of $\mathrm{fpt}(f)$ is always divisible by $p$ [1404.5871]. This resolves a question of Schwede and underscores the intricate tightness between the arithmetic of $p$ and the possible thresholds.

## 4. Syzygy Gap Fractals and Algorithmic Computation in Two Variables

A crucial combinatorial tool for understanding F-pure thresholds and all $F$-thresholds in the two-variable homogeneous case is the "syzygy gap fractal" function, defined for polynomials written as products of linear forms. This function encodes the F-threshold function for any collection of exponents, and its critical points correspond to sharp thresholds where the denominator in lowest terms is always a multiple of $p$ [1404.5871].

Main results include:
- When $\mathrm{fpt}(f) < 2/d$ (the "expected" value), its denominator is necessarily divisible by $p$.
- An explicit and efficient algorithm exists for computing F-pure thresholds in two variables, based on iterated colon ideals and base-$p$ digital expansions [1404.5871].

## 5. Refined Existence and Realization Results

The landscape of realizable F-pure thresholds is highly arithmetic and sensitive to both $d$ and $p$. Smith–Vraciu establish:
- For every integer $d \geq 4$ not divisible by $p$, there exists a reduced homogeneous polynomial whose FPT is a truncation of $2/d$ at the place $e = \operatorname{ord}_{(\mathbb{Z}/d)^{\times}}(p)$ [2207.11800].
- Low-degree cases ($d \leq 8$) admit complete characterization: all FPTs are precisely those predicted by the truncation and order criterion.

The aforementioned counterexample for $d = 2p-3$ (with $p > 7$) demonstrates that additional congruence conditions, as previously believed, are insufficient [2207.11800].

## 6. Open Problems and Future Directions

Major open questions concern:
- The full stratification of the parameter space of degree-$d$ forms by F-pure threshold: for $n \geq 3$, existence of reduced forms on every stratum remains unknown.
- A necessary and sufficient combinatorial description of all possible truncations that yield F-pure thresholds for reduced forms, beyond order and congruence constraints.
- Further exploration of connections between F-pure thresholds, Hilbert–Kunz multiplicity, and other Frobenius-splitting invariants.

The refined truncation approach of Smith–Vraciu, algorithmic advances via syzygy gap fractals, and the structural denominators result of Hernández–Núñez-Betancourt–Witt–Zhang collectively provide a comprehensive modern arithmetic theory of the F-pure threshold for homogeneous and binomial hypersurfaces in prime characteristic [2207.11800, 1404.5871, 1404.3772].

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**References**:  
[2207.11800] Smith and Vraciu: "Values of the F-pure threshold for homogeneous polynomials"  
[1404.5871] Hernández, Teixeira, Witt, Zhang: "F-threshold functions: syzygy gap fractals and the two-variable homogeneous case"  
[1404.3772] Hernández, Núñez-Betancourt, Witt, Zhang: "F-pure thresholds of homogeneous polynomials"  
[1112.2427] Hernández: "F-pure thresholds of binomial hypersurfaces"  
[2506.18891] Baily: "On lower bounds for the F-pure threshold of equigenerated ideals"  
[2501.13613] Blickle, Schwede, Tucker: "The defect of the F-pure threshold"  
[1106.0207] Bhatt, Hernández, Miller, Mustaţă: "Log canonical thresholds, F-pure thresholds, and non-standard extensions"

Source: https://www.emergentmind.com/topics/f-pure-threshold-fpt