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On $F$ -thresholds of differential power filtrations

This paper establishes the existence and provides an upper bound for the $F$ -thresholds of differential power filtrations of monomial ideals, demonstrating that for the monomial maximal ideal, this threshold equals the dimension of the polynomial ring.

Original authors: Wágner Badilla-Céspedes

Published 2026-07-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Wágner Badilla-Céspedes

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to measure how "rough" or "bumpy" a mathematical landscape is. In the world of algebra, these landscapes are built from shapes called ideals, and the bumps are singularities (places where the math gets messy).

For a long time, mathematicians have used a special ruler called an F-threshold to measure these bumps. Think of this ruler as a way to compare two different ways of building walls:

Ordinary Powers: Just stacking bricks on top of each other.
Frobenius Powers: A magical, high-speed construction method that only works in a specific type of universe (one with a "prime characteristic," which is just a fancy way of saying the math behaves like a clock that resets after a certain number of ticks).

The F-threshold is the point where the magical wall finally catches up to the ordinary wall. If the magical wall is too slow, the number is small; if it's fast, the number is big. This number tells us a lot about the shape of the landscape.

The New Tool: Differential Power Filtrations

Recently, a mathematician named Koley and Kumar introduced a new way to build these walls, called filtrations. Instead of just one wall, imagine a whole series of walls getting thicker and thicker. They proved that you can measure the F-threshold for these series, but only if the series follows certain rules.

Now, enter Wágner Badilla-Céspedes, the author of this paper. They are looking at a very specific, tricky type of wall-building called differential power filtrations.

Here's the twist: In some math worlds (specifically those with "perfect fields"), these differential walls are exactly the same as another famous type of wall called symbolic powers. But in other worlds (where the field isn't perfect), they are not the same. In fact, the paper explicitly points out that if you assume they are always the same, you might get the wrong answer. For example, in a specific ring involving polynomials like $x^p - t$ , the differential wall of order 2 is different from the symbolic wall of order 2. So, we can't just copy-paste the old rules; we need to check these differential walls on their own.

The Big Discovery: The Monomial Case

The author focuses on a specific, manageable type of landscape: monomial ideals. You can think of these as shapes built only from single variables (like $x$ , $y$ , $z$ ) multiplied together, without any messy additions or subtractions.

The paper proves two main things:

The Number Exists: First, the author shows that for these monomial shapes, the F-threshold for differential power filtrations actually exists. It's not a number that runs off to infinity or behaves chaotically; it settles down to a specific value.
The Ceiling: The author finds a "ceiling" for this number. They prove that the F-threshold can never be higher than the maximum height of the minimal primes of the ideal.
- Analogy: Imagine the ideal is a castle. The "minimal primes" are the smallest, most essential towers that hold the castle up. The "height" is how many floors those towers have. The paper proves that the F-threshold (our roughness ruler) can never be taller than the tallest of these essential towers.

The paper doesn't just guess this; it proves it using a clever trick. They show that if you build a wall using a specific number of steps (related to the number of generators of the prime ideals), that wall will definitely be inside the "magical" wall (the Frobenius power). This forces the F-threshold to stay below that specific limit.

The Special Case: The Maximal Ideal

To show their new ruler works, the author tests it on the most famous shape of all: the monomial maximal ideal. This is the ideal generated by all the variables in the room ( $x_1, x_2, \dots, x_d$ ).

They prove that for this specific shape, the F-threshold of the differential power filtration is exactly equal to $d$ , which is the number of variables (or the dimension) of the polynomial ring.

Analogy: If you are in a 3D room (variables $x, y, z$ ), the "roughness" of the corner where all three walls meet is exactly 3. If you are in a 10-dimensional hyper-room, the roughness is 10.

What This Means (and What It Doesn't)

The paper proves that for monomial ideals, these F-thresholds exist and are bounded by the height of the minimal primes. It does not claim to solve the problem for all types of ideals (only monomial ones). It does not say that differential powers are always the same as symbolic powers (in fact, it highlights where they differ).

The author's work connects two different worlds: the world of differential operators (calculus-like tools) and the world of prime characteristic invariants (F-thresholds). By showing that the F-threshold exists for these specific differential walls, they give mathematicians a new, reliable tool to measure singularities in polynomial rings, provided those rings are built from monomials.

In short: We now know that for monomial ideals, the "roughness" of differential power filtrations is a real, calculable number, and it will never exceed the height of the smallest towers holding the ideal up. And for the simplest case of all, that roughness is exactly the number of dimensions in the space.

Technical Summary: On F-thresholds of Differential Power Filtrations

Problem Statement
The paper addresses the computation and existence of F-thresholds, a numerical invariant in commutative algebra and singularity theory used to measure the behavior of singularities in prime characteristic $p > 0$ . While F-thresholds are well-defined for ordinary powers of ideals (comparing $I^t$ against Frobenius powers $J^{[p^e]}$ ), recent work by Koley and Kumar extended this concept to filtrations of ideals, such as symbolic powers. However, the behavior of F-thresholds for differential power filtrations remains less explored. Differential powers, introduced by Dao et al., generalize symbolic powers but diverge from them in rings that are not over perfect fields. The central problem is to determine whether the F-threshold exists for the filtration of differential powers of monomial ideals and to establish explicit bounds for this value.

Methodology
The author employs techniques from the theory of differential operators in prime characteristic and the combinatorial properties of monomial ideals.

Definitions and Framework: The paper utilizes the definition of F-thresholds for filtrations $\mathfrak{a}_\bullet$ with respect to an ideal $J$ , defined via the limit of the ratio $\nu_J^{\mathfrak{a}_\bullet}(p^e)/p^e$ , where $\nu$ measures the largest power $t$ such that $\mathfrak{a}_t \not\subseteq J^{[p^e]}$ .
Differential Operators: The study relies on the ring of $K$ -linear differential operators $D_R$ and the definition of the $n$ -th differential power $I\langle n \rangle = \{f \in R \mid \delta(f) \in I \text{ for all } \delta \in D_R^{n-1}\}$ .
Monomial Case Analysis: The paper restricts the analysis to the polynomial ring $R = K[x_1, \dots, x_d]$ over a field of characteristic $p$ . It focuses on monomial ideals $I$ .
Key Lemma: A crucial step involves proving that for a monomial prime ideal $\mathfrak{p}$ , the differential power $\mathfrak{p}\langle n \rangle$ coincides with the ordinary power $\mathfrak{p}^n$ . Furthermore, the author establishes that for sufficiently large $N$ (specifically $N \geq \mu(\mathfrak{p})$ ), $\mathfrak{p}\langle N p^e \rangle \subseteq \mathfrak{p}^{[p^e]}$ .
Filtration Property: The paper proves that for any ideal $I$ in a polynomial ring, the sequence of differential powers $\{I\langle i \rangle\}_{i \geq 0}$ forms a valid filtration (satisfying $I\langle n \rangle I\langle m \rangle \subseteq I\langle n+m \rangle$ ), thereby allowing the application of Koley and Kumar's existence theorems.

Key Contributions and Results
The primary contribution is the establishment of the existence and an upper bound for the F-threshold of differential power filtrations of monomial ideals.

Theorem A (Theorem 3.6): Let $I \subseteq J \subseteq R$ be nonzero proper ideals, with $I$ being a monomial ideal and $J$ being a radical ideal. The F-threshold $C_J(I\langle \bullet \rangle)$ exists. Furthermore, it is upper bounded by the maximum height of the minimal primes of $I$ :
$C_J(I\langle \bullet \rangle) \leq \max\{\text{ht}(\mathfrak{p}) \mid \mathfrak{p} \in \text{Min}(I)\}.$
The proof relies on decomposing the monomial ideal into its minimal primes and utilizing the inclusion $\mathfrak{p}\langle N p^e \rangle \subseteq \mathfrak{p}^{[p^e]}$ to bound the growth of the filtration relative to the Frobenius powers.
Corollary 3.7: As a specific application, the paper calculates the F-threshold for the differential power filtration of the maximal ideal $\mathfrak{m} = (x_1, \dots, x_d)$ . It shows that $C_{\mathfrak{m}}(\mathfrak{m}\langle \bullet \rangle) = d$ , which coincides with the dimension of the polynomial ring. This is demonstrated by showing the lower bound matches the dimension, while the upper bound is provided by Theorem 3.6.

Significance
The paper situates differential power filtrations within the broader framework of F-thresholds, effectively connecting differential operator techniques with invariants in prime characteristic. By proving the existence of these thresholds for monomial ideals, the work extends the known landscape of computable F-thresholds beyond symbolic powers. The result that the F-threshold of the differential power filtration of the maximal ideal recovers the dimension of the ambient ring serves as a consistency check, aligning this new invariant with classical geometric expectations. The work provides a concrete upper bound dependent on the height of minimal primes, offering a tool for estimating these invariants in the monomial setting without requiring explicit computation of the limit for every case.

The New Tool: Differential Power Filtrations

The Big Discovery: The Monomial Case

The Special Case: The Maximal Ideal

What This Means (and What It Doesn't)

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