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Counting w-coprime S-integers and S-integral ideals in positive characteristic

This paper employs a combination of analytic methods, the Riemann-Roch theorem, and the Weil theorem to derive counting formulas for $w$ -coprime $S$ -integers and $S$ -integral ideals within an algebraic function field over a finite field of positive characteristic.

Original authors: Si-Han Liu, Zhe-Cheng Liu, Jia-Yan Yao

Published 2026-06-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Si-Han Liu, Zhe-Cheng Liu, Jia-Yan Yao

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: Counting "Special" Numbers in a Different World

Imagine you are playing a game with numbers, but not the usual numbers (1, 2, 3...) we use every day. Instead, you are playing in a "Finite World" called Positive Characteristic.

In our normal world, numbers go on forever. In this Finite World, everything is built from a small, fixed set of building blocks (like a deck of cards with only $q$ cards). Even though the world is finite, it has its own version of "arithmetic" that behaves surprisingly like our own.

The authors of this paper are trying to answer a very specific question: If you pick a bunch of these special numbers at random, how likely is it that they share no common "bad" factors?

The Core Concepts (Translated)

1. The "S-Integers": A VIP Club

Imagine the set of all numbers in this Finite World. Now, imagine there is a special "VIP Club" called $O_S$ .

The Rule: You can join this club if you don't have any "debt" (negative values) at specific locations, except for a few special locations allowed in the club's charter (the set $S$ ).
The Analogy: Think of the whole world as a city. The "S-integers" are the people who live in the city but are allowed to have "construction zones" (special places) where they can break the usual rules. Everywhere else, they must follow the rules perfectly.

2. "Coprime": The No-Common-Factor Rule

In normal math, two numbers are coprime if they don't share any common building blocks (factors) other than 1. For example, 8 and 9 are coprime (factors of 8 are 1,2,4,8; factors of 9 are 1,3,9). But 8 and 12 are not coprime because they both share the factor 4.

The paper looks at a more advanced version called $w$ -coprime.

The Analogy: Imagine you are checking a group of people to see if they all share a specific "bad habit."
- If $w=1$ , you check if they share any bad habit.
- If $w=2$ , you check if they share a bad habit that happens to be a "square" (like a square habit).
- If $w=3$ , you check for "cubes," and so on.
The Goal: The authors want to count how many groups of $m$ numbers are $w$ -coprime, meaning they don't all share a bad habit of the $w$ -th power type.

3. The Two Main Counts

The paper calculates the number of these special groups in two different ways:

Counting the Numbers (Theorems 1 & Corollary 1):
They count individual numbers (elements) in the VIP club ( $O_S$ ) that are $w$ -coprime.
- The Result: As the numbers get bigger, the probability that a random group of $m$ numbers is $w$ -coprime is exactly $1 / \zeta_S(wm)$ .
- What is $\zeta_S$ ? Think of this as a "complexity meter" for this specific VIP club. It's a special formula that sums up all the possible bad habits in the club. The formula tells us exactly how rare or common these "clean" groups are.
Counting the "Ideal" Groups (Theorems 2 & Corollary 3):
Instead of counting individual numbers, they count "Ideal Groups" (which are like bundles or packages of numbers).
- The Result: The probability that a random bundle of $m$ packages is $w$ -coprime is also $1 / \zeta_S(wm)$ .
- Why this matters: It shows that whether you look at the individual people or the groups they form, the "cleanliness" ratio is the same.

How They Did It (The Toolkit)

The authors didn't just guess; they used a powerful combination of three tools:

The Sieve (Inclusion-Exclusion):
Imagine you have a basket of eggs, and you want to remove all the cracked ones. You check for cracks of type A, then type B, then type C. But if you just subtract them, you might accidentally remove an egg that was cracked twice. The "Sieve" method is a clever way to count and subtract so you don't double-count or miss anything. The authors used a "Sieve" adapted for this Finite World.
The Riemann-Roch Theorem (The Counting Machine):
This is a famous rule in math that helps you count how many ways you can arrange things in a specific space.
- The Analogy: Imagine you have a garden (the Finite World) and you want to know how many different flower arrangements fit in a specific plot size. This theorem gives you a precise formula for that count, even if the garden is weirdly shaped. The authors used this to count how many numbers fit inside their "VIP Club" boundaries.
The Weil Theorem (The Error Checker):
When you make a big estimate, there's always a tiny bit of "noise" or error. The Weil theorem is like a high-precision ruler that tells the authors exactly how big that error can be.
- The Achievement: The authors managed to prove their results with a very small error margin (a "tight" estimate), which is better than some previous attempts in similar fields.

The "So What?" (Without the Jargon)

The Main Discovery: In this strange, finite mathematical world, the chance of picking a group of numbers that don't share a specific type of common factor is predictable. It follows a precise mathematical law involving the "complexity meter" ( $\zeta_S$ ).
The Connection to Reality: This is similar to a famous result in our world: If you pick two random numbers, there is a 60% chance ( $6/\pi^2$ ) they are coprime. This paper proves that a similar rule exists in this "Finite World" for much more complex groups of numbers and different types of "coprimality."

Summary

The paper is a mathematical detective story. The authors went into a finite, abstract world, built a special club ( $O_S$ ), and asked: "How many groups of members here are 'clean' (w-coprime)?" Using a mix of sieves, garden-counting rules, and error-checking rulers, they found the exact answer and proved that the "cleanliness" of these numbers follows a beautiful, predictable pattern.

Technical Summary: Counting $w$ -coprime $S$ -integers and $S$ -integral ideals in positive characteristic

Problem Statement
This work addresses the problem of counting $w$ -coprime tuples within the context of algebraic function fields over finite fields. Specifically, the authors investigate two related counting problems:

$w$ -coprime $S$ -integers: Counting $m$ -tuples of elements $(a_1, \dots, a_m)$ in the ring of $S$ -integers $O_S$ of a function field $K$ such that the ideal generated by the tuple is not contained in any prime power $\mathfrak{q}^w$ (where $\mathfrak{q}$ is a prime ideal in $O_S$ ), subject to constraints on the valuations at a finite set of prime divisors $S$ .
$w$ -coprime $S$ -integral ideals: Counting $m$ -tuples of integral ideals $(\mathfrak{a}_1, \dots, \mathfrak{a}_m)$ in $O_S$ satisfying the same $w$ -coprimality condition, subject to constraints on the degree of the ideals.

The concept of $w$ -coprimality generalizes standard coprimality (where $w=1$ ) and square-free conditions (where $w=2$ and $m=1$ ). The study is set in positive characteristic, where $K$ is an algebraic function field over a finite field $\mathbb{F}_q$ with field of constants $\mathbb{F}_q$ .

Methodology
The proofs rely on a synthesis of three primary mathematical tools:

Inclusion-Exclusion Principle: The authors utilize a Möbius inversion formula adapted to the group of divisors $\text{Div}_S$ (generated by prime divisors outside $S$ ). They define a Möbius function $\mu_S$ on divisors and ideals to isolate tuples that do not satisfy the "bad" property of being contained in $\mathfrak{q}^w$ .
Riemann-Roch Theorem: This theorem is employed to estimate the dimension $\ell(D)$ of the Riemann-Roch space $L(D)$ associated with specific divisors. This dimension determines the cardinality of the sets of $S$ -integers satisfying valuation constraints. The authors carefully handle cases where the degree of the divisor is small (less than $2g_K - 1$ ) versus large, using the genus $g_K$ of the function field.
Weil's Theorem and Zeta Functions: The asymptotic behavior of the counting functions is derived using the properties of the $S$ -zeta function $\zeta_S(s)$ . The authors leverage the analytic properties of the zeta function over function fields, specifically the pole structure and the polynomial nature of the numerator in the functional equation (Weil's theorem), to evaluate the main terms and bound the error terms.

Key Contributions and Results

Theorem 1 (Counting $w$ -coprime $S$ -integers):
The paper establishes an asymptotic formula for $V^w_m(\vec{N})$ , the number of $w$ -coprime $m$ -tuples in a specific bounded region defined by a vector $\vec{N}$ . The formula is:
$V^w_m(\vec{N}) = \frac{q^{m(N+1-g_K)}}{\zeta_S(wm)} + O_K(q^{N/w})$
where $N = \sum N_j \deg p_j$ , $g_K$ is the genus, and $\zeta_S$ is the $S$ -zeta function. The error term is explicitly bounded and depends on $K$ . The result holds for integers $w, m \ge 1$ with $wm \ge 2$ . The case $w=m=1$ is excluded as it corresponds to counting units, which follows a different asymptotic behavior.
Theorem 2 (Counting $w$ -coprime $S$ -integral ideals):
The authors provide an asymptotic formula for $Q^w_{S,m}(n)$ , the number of $w$ -coprime $m$ -tuples of integral ideals with degree $\le n$ . The main term is:
$Q^w_{S,m}(n) = \frac{c_{K,S}^m q^{mn}}{\zeta_S(wm)} + \text{Error Term}$
The error term varies based on $m$ and $w$ :
- $O_{K,S}(q^{n/w})$ if $m=1$ .
- $O_{K,S}(n q^n)$ if $m=2, w=1$ .
- $O_{K,S}(q^{n(m-1)})$ otherwise.
  Here, $c_{K,S}$ is a constant involving the class number $h_K$ and the degrees of primes in $S$ .
Corollary 2 (Schanuel's Theorem for Function Fields):
Applying Theorem 1, the authors derive a new proof for the asymptotic formula of the number of points in projective space $\mathbb{P}^n(K)$ with height $\le B$ . They obtain:
$N_K(B, n) = \frac{h_K q^{(n+1)(B-g_K+1)}}{(q-1)\zeta_K(n+1)(1-q^{-(n+1)})} + O_K(q^B)$
The authors note that while the error term $O(q^B)$ is worse than the $O(q^{B/2+\epsilon})$ achieved by Dipippo and Wan, it is an elementary proof that improves upon the error terms of Serre and Phillips.

Significance and Claims
The paper claims to provide results with "relatively better error term estimates" compared to some existing literature in the specific setting of general algebraic function fields in positive characteristic. The significance lies in:

Generalization: Extending known results on $w$ -coprime integers and ideals from number fields and polynomial rings to general function fields with a finite set of excluded primes ( $S$ -integers).
Explicit Error Terms: Providing explicit constants and error bounds derived from the genus and class number of the field, rather than relying on unproven hypotheses like the Extended Lindelöf Hypothesis (which was required for some number field results).
Elementary Proof of Schanuel's Theorem: Offering a new, elementary proof for the asymptotic count of points in projective space over function fields, distinct from the geometric number theory approaches used by Phillips or the earlier proofs by Dipippo and Wan.

The authors emphasize that the probability for $m$ randomly chosen elements (or ideals) to be $w$ -coprime converges to $1/\zeta_S(wm)$ , mirroring classical results in number theory but adapted to the function field context.