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Turning non-smooth points into rational points

This paper improves upon a previous bound by establishing a new, sharp bound on the number of iterated Frobenius pullbacks required to transform a non-smooth purely inseparable point on a regular geometrically integral curve into a rational point for every characteristic $p>0$ .

Original authors: Cesar Hilario

Published 2026-06-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Cesar Hilario

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

The Big Picture: Smoothing Out a Bumpy Road

Imagine you are driving a car on a road (a mathematical "curve"). In the perfect world of mathematics, some roads are perfectly smooth, while others have bumps, potholes, or jagged edges. In this paper, the author is studying a specific type of "bump" on a road that exists only in a strange, specific type of mathematical universe (characteristics $p > 0$ , which is like a world where numbers wrap around after a certain point, similar to a clock).

These bumps are called non-smooth points. They are tricky because, in their current state, they don't behave nicely. You can't easily "park" your car (find a rational point) exactly on them.

The author's goal is to answer a simple question: How many times do we need to "zoom in" or "transform" the road before that bumpy spot becomes a smooth, parkable spot?

The Magic Tool: The Frobenius Pullback

To fix the road, the author uses a special tool called the Frobenius pullback. Think of this as a magical lens or a time-travel machine.

The Process: Every time you use this lens, the road changes slightly. The "bump" (the non-smooth point) gets stretched and shifted.
The Result: If you keep using this lens over and over again, eventually, the bump disappears. The point becomes smooth and rational.
- Smooth means the road is flat there.
- Rational means you can actually park your car there (it's a point with coordinates that make sense in the original language of the road).

The paper asks: What is the exact number of times ( $n$ ) we need to use this lens to guarantee the bump is gone?

The Old Map vs. The New Map

In a previous paper (by the author and a colleague named Stöhr), they had a map that told them roughly how many times to use the lens.

The Old Map: It worked perfectly for some situations (like when the "clock" only has 2 hours, i.e., characteristic $p=2$ ). But for other situations (like clocks with 3, 5, or 7 hours), the map was a bit vague. It gave a safe upper limit, but it wasn't the tightest possible limit. It was like saying, "You might need to drive 100 miles to get there," when sometimes 50 miles is enough.
The New Map (This Paper): The author has created a sharp, precise map.
- They figured out the exact minimum number of times you need to use the lens for every possible type of bump.
- They call this number $\lambda_p(d)$ .
- "Sharp" means the map is perfect. If the map says you need 5 zooms, it is impossible to do it in 4. There is no wiggle room.

The "Bump" Size Matters

The paper explains that not all bumps are the same. Some are tiny, some are huge. The author categorizes them by a number called the singularity degree (let's call it the "bumpiness score").

The Rule: The bigger the bumpiness score, the more times you might need to use the lens.
The Discovery: The author found a specific formula to calculate the exact number of lens uses needed based on the bumpiness score and the type of "clock" (the characteristic $p$ ) you are working in.

How They Proved It: Building the Worst-Case Scenarios

To prove their new map is the absolute best, the author didn't just guess. They played a game of "What if?"

Constructing the Impossible Road: They built specific, imaginary mathematical roads designed to be as stubborn as possible. These roads have bumps that refuse to become smooth until you hit the exact number of steps the author predicted.
The Test: They showed that for these specific stubborn roads, if you stop one step too early, the bump is still there.
The Conclusion: Since they found a road that requires exactly $X$ steps, and they already knew no road ever needs more than $X$ steps, they proved that $X$ is the perfect, unbreakable limit.

A Note on the "Language" of the Road

The paper also touches on a subtle detail: the "language" the road speaks.

Sometimes the road speaks a "pure" language (purely inseparable).
Sometimes it speaks a mix of pure and "separable" languages.

The author shows that their new sharp map works for the pure language cases. They also show that for the mixed language cases, the same logic applies, just with a slight adjustment for how "mixed" the language is.

Summary

The Problem: Bumpy points on mathematical roads that are hard to pin down.
The Solution: A magical process (Frobenius pullback) that eventually smooths them out.
The Contribution: The author calculated the exact number of times you must apply this process to guarantee the point is fixed.
Why it matters: Before this, we had a "safe estimate." Now, we have the "precise truth." This helps mathematicians understand the structure of these strange, bumpy roads in positive characteristic worlds much better, allowing them to classify and describe these curves with total accuracy.

The paper is essentially a masterclass in precision: taking a rough estimate and refining it into a perfect, unbreakable rule for every possible scenario.

Technical Summary: Turning Non-Smooth Points into Rational Points

Problem Statement
This paper addresses the behavior of non-smooth points on regular curves over imperfect fields of characteristic $p > 0$ . While regularity and smoothness coincide over perfect fields, they diverge in the imperfect setting, giving rise to regular curves that are non-smooth (e.g., quasi-elliptic curves). The central problem is to determine the precise number of iterated Frobenius pullbacks required to transform a non-smooth, purely inseparable point $x$ on a regular curve $C$ into a $K$ -rational point.

Previous work by the author and Stöhr [HS25a] established an upper bound, denoted $\tau_p(d)$ , for the number of pullbacks needed, where $d$ is a parameter related to the geometric singularity degree $\delta(x) = \frac{p-1}{2}d$ . However, this bound was known to be sharp only in characteristic $p=2$ . In characteristics $p > 2$ , the bound was not always optimal due to constraints on which singularity degrees can actually occur for purely inseparable points.

Methodology
The paper employs the duality between regular curves and one-dimensional separable function fields to analyze the problem arithmetically. The core methodology involves:

Admissibility Analysis: The authors first characterize which integers $d$ are "admissible," meaning there exists a pair $(C|K, x)$ with singularity degree $\delta(x) = \frac{p-1}{2}d$ . They prove that $d$ is admissible if and only if $d \not\equiv -1 \pmod p$ .
Construction of Extremal Examples: To establish sharp lower bounds, the authors explicitly construct specific function fields $F|K$ and purely inseparable primes $p$ with prescribed singularity degrees. These constructions utilize the Bedoya–Stöhr algorithm [BS87] and carefully chosen imperfect base fields (sometimes requiring $p$ -degree $\ge 2$ ).
Recursive Singularity Calculation: Using the Bedoya–Stöhr algorithm and Theorem 2.3, the authors compute the sequence of singularity degrees $\delta(p_n)$ for the restricted primes under iterated Frobenius pullbacks. They analyze the "differences" $\Delta_n = \delta(p_n) - \delta(p_{n+1})$ to determine exactly when the restricted prime becomes rational.
Combinatorial Optimization: The sharp bound $\lambda_p(d)$ is derived by finding the minimal $m$ such that for all admissible pairs, the restricted prime is rational for all $n \ge m$ . This involves comparing the constructed lower bounds against the existing upper bound $\tau_p(d)$ .

Key Contributions and Results

Sharp Bound $\lambda_p(d)$ : The primary result is Theorem 1.2 (and Theorem 3.5), which establishes a new, sharp bound $\lambda_p(d)$ for the number of Frobenius pullbacks needed to turn a non-smooth purely inseparable point into a rational point. This bound is optimal for every characteristic $p > 0$ and every admissible integer $d$ .
- The bound $\lambda_p(d)$ is either equal to the previous bound $\tau_p(d)$ or $\tau_p(d) - 1$ .
- The specific value depends on the $p$ -adic expansion of $d$ and whether $d$ can be represented as a sum of consecutive $p$ -powers in specific configurations.
- Corollary 3.6 confirms that in characteristic $p=2$ , the previous bound $\tau_p(d)$ is always sharp.
Generalization to Arbitrary Non-Smooth Points: Theorem 1.6 extends these results to arbitrary non-smooth points (not necessarily purely inseparable). It establishes that if a point $x$ has singularity degree $\delta(x) = \frac{p-1}{2} [\kappa(x):K]_s \cdot d$ , then its image becomes a separable point (lying in $C_n(K^{sep})$ ) for all $n \ge \lambda_p(d)$ . This generalizes the purely inseparable case where the separable degree is 1.
Admissibility Criterion: Proposition 3.2 provides a necessary and sufficient condition for an integer $d$ to be the singularity degree parameter of a purely inseparable point: $d \not\equiv -1 \pmod p$ .
Explicit Constructions: Section 4 provides explicit examples of function fields (Examples 4.1 and 4.2) that realize the sharp bounds. These examples demonstrate that the bound cannot be lowered further, as there exist pairs where the point remains non-rational (or non-separable) exactly until step $\lambda_p(d) - 1$ .

Significance and Claims
The paper claims to resolve the question of sharpness for the bound on Frobenius pullbacks in all characteristics $p > 0$ . While the previous bound $\tau_p(d)$ was known to be sharp for $p=2$ , this work demonstrates that for $p > 2$ , the bound can often be improved by 1, depending on the arithmetic properties of the singularity degree.

The authors note that their constructed examples share the property of being geometrically rational (genus 0). They state that while they do not have a priori control over the global geometry of these curves in general, the sharp bounds $\lambda_p(d)$ are expected to hold even under global restrictions, such as fixing the geometric genus or the $p$ -degree of the base field. The results provide a method for obtaining explicit descriptions of regular non-smooth curves by working backwards from the rational curve obtained after sufficient Frobenius pullbacks, a technique previously applied to quasi-elliptic curves and plane quartics.

The paper does not claim to solve the classification of all such curves but rather provides the precise arithmetic threshold required to make the "turning non-smooth points into rational points" strategy effective for any given singularity degree.