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Factorizations of linearized polynomials and extremal curves in odd characteristic

This paper presents a complete construction recipe for extremal van der Geer--van der Vlugt curves over finite fields of odd characteristic, utilizing a nonzero field element and a linear subspace satisfying a specific trace condition as input data.

Original authors: Tetsushi Ito, Daichi Takeuchi, Takahiro Tsushima

Published 2026-06-18

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CC BY 4.0

Original authors: Tetsushi Ito, Daichi Takeuchi, Takahiro Tsushima

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 an architect trying to build the most efficient structures possible in a very specific, mathematical universe called "finite fields." In this universe, numbers don't go on forever like 1, 2, 3...; instead, they wrap around like the hours on a clock.

This paper is a blueprint for building a special type of mathematical structure called a curve. Specifically, the authors are building "extremal" curves. Think of an extremal curve as a building that is either packed with the maximum possible number of rooms (points) or has the absolute minimum number of rooms, given its size (genus). These "perfectly packed" or "perfectly empty" buildings are rare and valuable in the world of mathematics, particularly for things like error-correcting codes (which keep your data safe) and cryptography.

Here is the simple breakdown of what the authors did:

1. The Problem: Finding the "Perfect" Curve

In the world of these finite fields, there is a known family of curves called van der Geer–van der Vlugt curves. They are defined by a specific equation involving a polynomial (a mathematical recipe with variables and exponents).

The authors wanted to answer a simple question: Which of these recipes produce the "perfect" (extremal) curves?

Previously, they had figured this out for a universe where the "clock" only had two hours (characteristic two). This paper solves the puzzle for universes where the clock has an odd number of hours (odd characteristic), like 3, 5, 7, etc.

2. The Key Ingredient: The "Adjoint" Mirror

To solve this, the authors use a clever trick involving a concept they call the adjoint.

Imagine you have a machine (a polynomial) that takes a number and transforms it. The "adjoint" is like a mirror image of that machine. If you run a number through the original machine and then through the mirror machine, you get a specific result.

The authors discovered a secret recipe:

If you take a specific type of machine (a linearized polynomial), create its mirror image, and combine them in a specific way, the result reveals the "fingerprint" of the curve.
This fingerprint tells you immediately if the curve is "extremal" (perfectly packed or empty) without having to count every single room in the building.

3. The Recipe for Construction

The paper provides a step-by-step "cookbook" for anyone who wants to build these perfect curves. You don't need to guess; you just need two ingredients:

A Subspace (W): Think of this as a specific, smaller group of numbers within your universe that follows certain rules.
A Constant (a): A single non-zero number that acts as a tuning knob.

The Magic Condition:
The authors found that if you pick your group of numbers and your tuning knob correctly, the curve will be perfect. The rule is surprisingly simple:

If you take any number from your group, square it, and mix it with your tuning knob, the "trace" (a specific mathematical sum) must equal zero.

If this condition is met, you have built an extremal curve. If not, you haven't.

4. The "Odd" vs. "Even" Difference

The authors note that building these curves in an "odd" universe (like a 3-hour clock) is slightly different from the "even" universe (a 2-hour clock) they studied before.

In the even universe, the rules were a bit more complex, requiring an extra parameter to handle the math.
In the odd universe, the rules are cleaner. The "mirror" trick works more directly, allowing them to create a complete list of every possible perfect curve in this family.

5. The Big Discovery: "All Roads Lead to Rome"

Perhaps the most important finding is that every perfect curve in this family can be built using their recipe.

Before this paper, mathematicians knew some ways to build these curves, but they didn't know if there were other hidden ways.
The authors proved that there are no hidden ways. If you find a perfect curve, it must have been built using their specific combination of a subspace and a tuning knob. It's like saying, "Every perfect cake in this bakery was made using this specific recipe; there are no secret recipes."

6. The Connection to the "Hermitian" Curve

Finally, the paper shows that these perfect curves are related to a famous, giant curve called the Hermitian curve.

Think of the Hermitian curve as a massive, master blueprint.
The authors prove that every "perfect" curve they built is essentially a smaller, simplified version (a quotient) of this giant master blueprint.
This means that if you understand the giant Hermitian curve, you automatically understand all the smaller perfect curves they constructed.

Summary

In short, this paper is a complete instruction manual for building the most efficient mathematical structures possible in odd-numbered finite fields. It provides a simple test to check if a structure is perfect, proves that this method finds all such structures, and connects them all to a famous mathematical giant. It's a "recipe book" that ensures no perfect curve is left behind.

Technical Summary: Factorizations of Linearized Polynomials and Extremal Curves in Odd Characteristic

Problem Statement
The paper addresses the construction and classification of extremal van der Geer–van der Vlugt curves over finite fields $\mathbb{F}_q$ of odd characteristic. These curves are smooth compactifications of affine curves defined by the equation $y^p - y = xR(x)$ , where $R(x)$ is a nonzero $\mathbb{F}_p$ -linearized polynomial. A curve is termed $\mathbb{F}_q$ -extremal if it is either $\mathbb{F}_q$ -maximal or $\mathbb{F}_q$ -minimal, meaning the number of its rational points attains the Hasse–Weil upper or lower bound, respectively. While a complete construction recipe for such curves in characteristic two was previously established by the authors, the case of odd characteristic remained less explicit. The primary objective is to provide a complete recipe for constructing these extremal curves in odd characteristic and to characterize the set of all such curves.

Methodology
The authors employ the theory of $\mathbb{F}_p$ -linearized polynomials and their adjoints to derive explicit formulas for the $L$ -polynomials of these curves. The core methodological innovation involves the notion of the adjoint of a linearized polynomial. For a polynomial $f(x) = \sum b_i x^{p^i}$ , the adjoint is defined as $f^*(x) = \sum (b_i x)^{p^{-i}}$ .

The construction proceeds as follows:

Input Data: The construction begins with a pair $\rho = (W, a)$ , where $W \subset \mathbb{F}_q$ is an $\mathbb{F}_p$ -linear subspace and $a \in \mathbb{F}_q^\times$ .
Polynomial Construction: From $W$ , a specific separable $\mathbb{F}_p$ -linearized polynomial $F_W(x)$ is constructed. The authors utilize a key factorization identity:
$R + R^* = F^* \circ (aF)$
where $F = F_W$ . This factorization uniquely determines a polynomial $R_\rho(x)$ satisfying the condition.
Curve Definition: The curve $D_\rho$ is defined by $y^p - y = xR_\rho(x)$ .
L-Polynomial Computation: Using étale cohomology, Artin–Schreier sheaves, and the Grothendieck–Lefschetz trace formula, the authors compute the $L$ -polynomial of $D_\rho$ . This involves evaluating exponential sums related to quadratic forms over finite fields.
Extremality Criteria: By analyzing the eigenvalues of the Frobenius action derived from the $L$ -polynomial, the authors establish necessary and sufficient conditions for $D_\rho$ to be extremal.

Key Contributions and Results

Explicit Construction Recipe (Theorem 1.2): The paper provides a complete characterization of $\mathbb{F}_q$ $F_{q}$ -extremal van der Geer–van der Vlugt curves in odd characteristic. A curve $D_\rho$ $D_{ρ}$ associated with a pair $\rho=(W, a)$ $ρ = (W, a)$ is extremal if and only if the trace condition $\text{Tr}_{q/p}(a^{-1}v^2) = 0$ $Tr_{q / p} (a^{- 1} v^{2}) = 0$ holds for all $v \in W$ $v \in W$ .
- Furthermore, the paper distinguishes between maximal and minimal cases based on the Legendre symbol: $D_\rho$ is maximal (resp. minimal) if and only if $\left(\frac{a}{q}\right) \cdot \left(\frac{-1}{\sqrt{q}}\right) = -1$ (resp. $= 1$ ).
Completeness of the Construction (Theorem 1.2(ii)): A significant result is the proof that every $\mathbb{F}_q$ -extremal van der Geer–van der Vlugt curve arises from such a pair $(W, a)$ . This establishes that the constructed family exhausts the set of extremal curves within this class.
L-Polynomial Formula (Theorem 1.1): An explicit formula for the $L$ -polynomial is derived, expressed as a product over characters and elements of the subspace $W$ . This formula allows for the direct determination of the number of rational points.
Comparison with Characteristic Two: The paper highlights the structural similarities and differences between the odd and even characteristic cases. In odd characteristic, the condition for extremality relates to a quadratic form being totally isotropic, whereas in characteristic two, it involves a group homomorphism condition on the subspace.
Applications to Quotients and Hermitian Curves:
- The authors prove that for any prescribed genus within the allowable range, there exist both $\mathbb{F}_q$ -maximal and $\mathbb{F}_q$ -minimal curves in this family (Theorem 4.6).
- They demonstrate that any $\mathbb{F}_q$ -maximal van der Geer–van der Vlugt curve is a finite étale quotient of the Hermitian curve (Theorem 4.10). This is achieved by iteratively constructing curves of increasing genus that cover the original curve, eventually reaching the maximal degree corresponding to the Hermitian curve.
Equivalence to Previous Work: The paper clarifies that their extremality condition is equivalent to a criterion previously established in [6], but their approach offers a more constructive framework that proves the surjectivity of the construction (i.e., that all extremal curves are captured by this method).

Significance
The paper claims its primary significance lies in providing a "complete recipe" for constructing extremal curves in odd characteristic, filling a gap left by previous studies that focused on characteristic two or provided non-constructive criteria. By linking the geometry of these curves to the factorization of linearized polynomials and the theory of quadratic forms, the authors unify the treatment of maximal and minimal curves. The result that every extremal curve in this family is a quotient of the Hermitian curve reinforces the central role of the Hermitian curve in the geometry of maximal curves over finite fields. The work serves as an odd-characteristic analogue to the authors' previous results in characteristic two, offering a parallel theoretical framework for number theorists and coding theorists studying these curves.