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Classification of Rational Functions of Degree Three over Finite Fields

This paper completes the classification of rational functions of degree three over finite fields in odd characteristic by analyzing value frequencies and ramification points, thereby extending previous results that covered only even characteristic.

Original authors: Xiang-dong Hou, Siyu Peng, Yongyu Qiang, Shujun Zhao

Published 2026-04-22

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CC BY 4.0

Original authors: Xiang-dong Hou, Siyu Peng, Yongyu Qiang, Shujun Zhao

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 master architect working in a magical city called Finite Field. In this city, the laws of math are slightly different: there are only a fixed number of buildings (numbers), and if you keep counting, you eventually wrap around and start over.

Your job is to design Rational Functions. Think of these functions as magic elevators. They take a person standing on a specific floor (a number in the city) and transport them to a new floor. Some elevators are simple (like going up one floor), but your specialty is Degree 3 elevators. These are complex machines that can take three different starting floors and, in some cases, drop them all off at the exact same destination floor.

The Big Problem: Sorting the Elevators

The city is full of these elevators. But here's the catch: many elevators look different on the outside but work exactly the same on the inside.

For example, if you take an elevator, paint it blue, and rename the buttons, it's still the same machine. In math, we call this Equivalence. Two elevators are "equivalent" if you can just rename the floors (using a simple, linear transformation) before and after the ride, and the result is the same.

The big question the authors asked was: "How many truly unique, non-equivalent Degree 3 elevators exist in this city?"

Previously, mathematicians had solved this for cities with an even number of floors. But for cities with an odd number of floors (like 3, 5, 7, 9...), it was a mess. No one knew the full list. This paper is the ultimate catalog that finally sorts them all out.

The Three Types of Elevators

To solve this, the authors divided the elevators into three distinct neighborhoods based on how they behave:

1. The "One-to-One" Neighborhood (Class I)

These are the perfect elevators. No matter which floor you start on, you never land on the same floor as someone else. They are "permutations."

Analogy: Imagine a dance where everyone has a unique partner. No two people ever share a partner.
Status: These were already well-known. The authors just listed them in a table.

2. The "Double-Booked" Neighborhood (Class II)

In this neighborhood, there is at least one destination floor where exactly two different starting floors arrive.

Analogy: Imagine a bus stop where two different people get off at the same time, but everyone else gets off alone.
The Solution: The authors found a "standard blueprint" for these. They showed that every elevator in this neighborhood is just a slightly tweaked version of two basic designs. They proved that if you tweak the blueprint enough, you get a unique elevator.

3. The "Triple-Booked" Neighborhood (Class III) - The Hard Part

This is the tricky neighborhood. Here, there is a destination floor where three different starting floors all arrive. However, crucially, there is no floor where exactly two people arrive.

Analogy: Imagine a magic trick where three distinct people vanish and reappear at the same spot, but no two people ever share a spot.
The Challenge: This is where the math gets really hard. In even-numbered cities, these elevators are simple. But in odd-numbered cities, they can have up to four "weak spots" (called ramification points). These are the floors where the elevator's mechanism gets "stuck" or behaves strangely.

The Secret Weapon: The "Cross Ratio" and "Invariants"

How did the authors tell these complex elevators apart? They used a mathematical tool called the Cross Ratio.

The Metaphor: Imagine you have four special landmarks on a map. The "Cross Ratio" is like a unique fingerprint that describes the geometric relationship between these four points. No matter how you rotate or stretch the map (as long as you don't tear it), this fingerprint stays the same.
The Innovation: The authors created a new "fingerprint" called $\theta$ (Theta).
- If two elevators have the same $\theta$ , they are likely the same elevator.
- For the most complex sub-group (Class III-b), $\theta$ wasn't enough. They had to invent a second fingerprint, $\lambda$ (Lambda), and eventually a third, $\mu^2$ .

Think of it like identifying a suspect.

$\theta$ is like checking their height. (Good, but many people are the same height).
$\lambda$ is like checking their shoe size. (Better).
$\mu^2$ is like checking their DNA. (Perfect match).

By combining these fingerprints, the authors could prove that they had found every single unique elevator in the city.

The "Computer vs. Human" Dance

The authors didn't just do this with pen and paper. The math got so complicated (involving huge polynomials and curves) that they had to use Mathematica (a powerful computer algebra system) to verify their work.

They wrote code to check if their formulas actually worked.
They used a famous mathematical rule called the Hasse-Weil bound (which is like a "speed limit" for how many points a curve can have) to prove that they hadn't missed any elevators.

The Grand Conclusion

The paper concludes with a complete list.

If the city has $q$ floors, there are exactly $q-1$ unique "Triple-Booked" elevators.
They provided a recipe (a table of formulas) for how to build every single one of them.

Why Does This Matter?

You might ask, "Who cares about magic elevators in a tiny city?"

Cryptography: These functions are used to build secure codes that protect your bank data and online passwords.
Error Correction: They help send data across space (like from Mars rovers) without errors.
Pure Beauty: It's like solving a massive jigsaw puzzle that no one knew how to finish. The authors didn't just find the pieces; they showed exactly how they fit together to form a perfect picture.

In short, this paper is the definitive encyclopedia for a specific type of mathematical machine, turning a chaotic mess of possibilities into a neat, organized, and perfectly understood collection.

1. Problem Statement

The paper addresses the classification of non-constant rational functions $f \in \mathbb{F}_q(X)$ of degree $n=3$ over a finite field $\mathbb{F}_q$ under PGL-equivalence.
Two rational functions $f$ and $g$ are defined as equivalent ( $f \sim g$ ) if there exist $\psi, \phi \in \text{PGL}(2, \mathbb{F}_q)$ such that $g = \psi \circ f \circ \phi$ .

Context: While the classification for degree 1 (trivial) and degree 2 is well-known, and degree 3 in even characteristic was recently solved by Mattarei and Pizzato (2024), the case for odd characteristic remained open.
Goal: To provide a complete classification of all degree 3 rational functions over $\mathbb{F}_q$ where $q$ is odd, determining the number of equivalence classes and providing explicit representatives for each class.

2. Methodology

The authors employ a structural approach based on the value frequencies of the rational function and the ramification points of the associated function field extension $\mathbb{F}_q(X)/\mathbb{F}_q(f)$ .

A. Classification into Three Main Classes

The authors partition the set of degree 3 rational functions into three disjoint classes based on the size of the pre-image $|f^{-1}(\alpha)|$ for $\alpha \in \mathbb{P}^1(\mathbb{F}_q)$ :

Class I: Functions where $|f^{-1}(\alpha)| = 1$ for all $\alpha$ . These are the permutation rational functions (already classified by Ferraguti and Micheli).
Class II: Functions where $|f^{-1}(\alpha)| = 2$ for some $\alpha$ .
Class III: The remaining case, where $|f^{-1}(\alpha)| = 3$ for some $\alpha$ , but $|f^{-1}(\beta)| \neq 2$ for any $\beta$ .

B. Canonical Forms and Invariants

Class I & II: The authors provide canonical forms and prove that these forms represent distinct equivalence classes.
- Class II representatives are of the form $\frac{X^3+t}{X}$ and $\frac{X^3+X^2+t}{X}$ .
Class III (The Core Contribution):
- The authors assume $f^{-1}(\infty) = \{0, 1, \infty\}$ via equivalence, reducing the function to the form:
  $f_{s,t}(X) = \frac{X^3 + sX + t}{X(X-1)}$
- The classification relies on the factorization of the derivative numerator polynomial $G_{s,t}(X) = X^4 - 2X^3 - sX^2 - 2tX + t$ .
- Ramification Types: Based on the factorization of $G_{s,t}$ over $\mathbb{F}_q$ and the characteristic $p$ , Class III is subdivided into four subclasses (III-a, III-b, III-c, III-d).
- Invariants: To distinguish equivalence classes within these subclasses, the authors construct algebraic invariants:
  1. $\theta(s,t)$ : Defined as $\frac{s^3}{t(1+s+t)}$ . This invariant is derived from the cross-ratio of ramification points and the structure of the function field.
  2. $\lambda$ : A secondary invariant based on the cross-ratio of the four ramification points, used specifically for Class III-b where $\theta$ alone is insufficient.
  3. $\mu^2$ : A stronger invariant for Class III-b, derived from the coefficients of the irreducible quadratic factors of $G_{s,t}$ .

C. Analytical Tools

Hasse-Weil Bound: Used to estimate the number of rational points on algebraic curves defined by the conditions of the invariants, ensuring that the constructed sets of representatives cover the necessary number of equivalence classes.
Cross-Ratio Theory: Utilized to analyze the action of $\text{PGL}(2, \mathbb{F}_q)$ on the set of ramification points.
Resultants and Discriminants: Used to determine the irreducibility and factorization types of $G_{s,t}$ .

3. Key Contributions and Results

A. Complete Classification for Odd Characteristic

The paper establishes that for odd $q$ , the total number of equivalence classes of degree 3 rational functions is $N(q,3) = q-1$ (for Class III) plus the counts for Classes I and II. The authors prove that their constructed sets of representatives are pairwise nonequivalent and cover the entire space.

B. Detailed Subclassification of Class III ( $p \ge 5$ )

The authors provide a complete breakdown of Class III based on the characteristic $p$ and the congruence of $q \pmod{12}$ :

Class III-a ( $G_{s,t}$ is irreducible):
- Ramification type: $(2/4, 2/4, 2/4, 2/4)$ .
- Distinguished by the invariant $\theta$ .
- Number of classes: $(q-1)/2$ .
- $\Theta_a = \{27 + a \mid a \text{ is a nonsquare in } \mathbb{F}_q\}$ .
Class III-b ( $G_{s,t}$ is a product of two distinct irreducible quadratics):
- Ramification type: $(2/2, 2/2, 2/2, 2/2)$ .
- Distinguished by the pair $(\theta, \lambda)$ or the stronger invariant $\mu^2$ .
- Number of classes: $(q-5)/2$ .
- The paper resolves a subtle case where $\theta$ and $\lambda$ fail to distinguish two classes (when $q \equiv \pm 1 \pmod{12}$ and specific values occur), proving that $\mu^2$ is the complete invariant.
Class III-c ( $G_{s,t}$ is a square of an irreducible quadratic):
- Ramification type: $(3/2, 3/2)$ .
- Exists only if $q \not\equiv 1 \pmod 6$ .
- Represented by a single class (if it exists) with $\theta = 27$ .
Class III-d ( $G_{s,t}$ has roots in $\mathbb{F}_q$ ):
- Ramification types vary: $(2/2, 2/2, 3/1)$ or $(3/1, 3/1)$ .
- Includes the case $f_{-3,1}$ (where $\theta=27$ ).
- Number of classes: 1 or 2 depending on $q$ .

C. Characteristic $p=3$

The authors adapt the classification for $p=3$ . Notably, Class III-c is empty in this case, and the counts for III-a and III-b are adjusted based on the properties of squares in characteristic 3.

D. Even Characteristic Comparison

While the main focus is odd characteristic, the paper includes an appendix comparing their results with the recent work of Mattarei and Pizzato for even characteristic, showing that the invariant $\theta$ suffices for the classification in that case as well.

4. Significance

Resolution of an Open Problem: This work completes the classification of degree 3 rational functions over finite fields, a fundamental problem in the theory of function fields and finite geometry.
Methodological Advancement: The paper demonstrates a powerful technique of combining ramification analysis with explicit invariant construction (using cross-ratios and resultants) to solve classification problems where direct enumeration is impossible.
Applications:
- Permutation Polynomials: The classification aids in understanding and constructing permutation rational functions, which are crucial in cryptography and coding theory.
- Irreducible Polynomials: Rational functions are used to construct high-degree irreducible polynomials; knowing the equivalence classes helps in optimizing these constructions.
- Function Field Theory: The results provide concrete examples of the interplay between the arithmetic of the base field and the geometry of the function field extension.

5. Conclusion

The authors successfully classify all degree 3 rational functions over finite fields of odd characteristic. By dividing the problem into three main classes and further subdividing Class III based on ramification types, they identified explicit invariants ( $\theta, \lambda, \mu^2$ ) that completely determine the equivalence classes. The work bridges the gap between theoretical bounds (Hasse-Weil) and explicit algebraic constructions, providing a definitive reference for future research in finite field arithmetic and applications.