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On Heights and Diameters of Ternary Cyclotomic and Inclusion-Exclusion Polynomials

The paper proves that every natural number occurs as the height of some ternary cyclotomic polynomial by demonstrating that for any odd prime $p$ and valid integer $h$ , there exist arbitrarily large primes $q$ and $r$ such that the height of $\Phi_{pqr}$ is exactly $h$ , while also providing specific values for its diameter.

Original authors: Gennady Bachman

Published 2026-02-10

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Gennady Bachman

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 Secret Life of Polynomials: A Guide to Heights and Diameters

Imagine you are an architect tasked with building a skyscraper using only specific, pre-cut blocks. These blocks are mathematical objects called Cyclotomic Polynomials.

In the world of math, these polynomials are like complex blueprints. When you expand them, they are made up of a long string of numbers called coefficients. For example, a polynomial might look like:
$1x^5 - 2x^4 + 3x^3 + 0x^2 - 1x + 1$

The numbers in front of the $x$ (the $1, -2, 3, 0, -1, 1$) are the "building blocks." This paper is essentially a study of the extremes of these numbers.

1. The Two Main Measurements: Height and Diameter

The author, Gennady Bachman, focuses on two specific ways to measure these "blueprints":

The Height ( $A(n)$ ): The Peak of the Mountain
Imagine the coefficients are like the elevation of a mountain range. Some numbers are positive (peaks), and some are negative (valleys). The Height is simply the single highest peak or the single deepest valley. If your coefficients are $\{1, -5, 3\}$ , your height is $5$.

The Big Question: Can we build a mountain range of any height we want? If I ask for a mountain exactly 100 units high, is there a polynomial that fits?

The Diameter ( $D(n)$ ): The Total Vertical Span
If the Height tells you how far you go from sea level, the Diameter tells you the total distance from the very bottom of the deepest valley to the very top of the highest peak. If your lowest point is $-5$ and your highest is $3$, the diameter is $8$.

2. The "Ternary" Team: The Rule of Three

The paper focuses on Ternary polynomials. In math, "ternary" means we are building these blueprints using exactly three prime numbers (like $3, 5,$ and $7$).

Think of these three primes as three different "tuning forks." When you strike them together, they create a specific vibration (the polynomial). The paper investigates how changing these three "tuning forks" changes the height and diameter of the resulting "vibration."

3. What did the author discover?

Before this paper, mathematicians knew that some heights were possible, but they weren't sure if every height was possible. It was like knowing you could build a 1-story house or a 10-story building, but not knowing if a 5-story building was mathematically "allowed."

The Breakthrough: "Yes, you can build anything!"
Bachman proved that if you pick a starting prime number, you can find other primes to pair with it to create a polynomial with any height you desire (up to a certain limit).

He essentially provided a "recipe book." He showed that if you want a height of $h$ , you can pick specific large primes $q$ and $r$ to act as your secondary and tertiary tuning forks, and the resulting "mountain range" will hit that exact height every single time.

The Diameter Discovery:
He also found that the "vertical span" (the diameter) isn't random. It follows a very strict pattern based on how the numbers relate to each other. He showed that the diameter is almost always either $2h$ or $2h-1$ . It’s like saying, "If you want a mountain of height $h$ , the total distance from valley to peak will always be roughly double that."

Summary in a Nutshell

If mathematicians are explorers mapping out a vast, infinite landscape of numbers, Bachman has just handed them a GPS and a Ruler.

He proved that the landscape isn't just random chaos; it is a structured world where you can precisely control how high your mountains go and how deep your valleys sink, simply by choosing the right "prime number ingredients."

This paper, titled "On Heights and Diameters of Ternary Cyclotomic and Inclusion-Exclusion Polynomials" by Gennady Bachman, addresses fundamental questions regarding the coefficient properties of cyclotomic polynomials, specifically focusing on the "ternary" case (polynomials with three distinct odd prime factors).

1. The Problem

The study of cyclotomic polynomials $\Phi_n(x)$ often centers on their coefficients $a(m; n)$ . Two specific metrics are of primary interest:

Height $A(n)$ : The maximum absolute value of the coefficients: $A(n) = \max_m |a(m; n)|$ .
Diameter $D(n)$ : The difference between the largest and smallest coefficients: $D(n) = \max_m a(m; n) - \min_m a(m; n)$ .

The central open question addressed is the solubility of the height equation: Given a natural number $h$ , does there exist an $n$ such that $A(n) = h$ ? While this is trivial for $k < 3$ (where $A(n)=1$ ), it becomes highly complex for $k \geq 3$ . Specifically, for ternary polynomials ($n=pqr$), the paper seeks to determine which heights $h$ are achievable for a fixed prime $p$ .

2. Methodology

The author employs a combinatorial approach by studying ternary inclusion-exclusion polynomials $Q_T(x)$ , where $T=\{p, q, r\}$ . When $p, q, r$ are distinct primes, $Q_T(x) = \Phi_{pqr}(x)$ .

Key technical tools include:

The Inclusion-Exclusion Principle: Expressing coefficients $a_m$ as sums of a characteristic function $\chi(n)$ , which indicates whether an integer $n$ can be represented as a non-negative linear combination of $qr, pr,$ and $pq$.
Modular Arithmetic Reductions: Using the Chinese Remainder Theorem and specific congruence conditions on $q$ and $r$ (relative to $p$ ) to simplify the search for coefficients.
Parameterization: The author introduces parameters $t$ and $s$ (inverses modulo $p$ ) to categorize the behavior of the coefficients. By choosing $q$ and $r$ such that they satisfy specific modular relations (e.g., $rt \equiv 1 \pmod{pq}$ ), the author can explicitly calculate the maximum and minimum coefficients.
Case Analysis of Sums: The proof relies on evaluating a specialized sum $S(Q, R; M)$ across three distinct cases of index ranges to determine the exact bounds of the coefficient set.

3. Key Contributions and Results

The paper provides several definitive results that advance the understanding of ternary cyclotomic polynomials:

A. Height Results (Theorem 2)

The author proves that for any odd prime $p$ and any integer $h$ in the range $1 \leq h \leq (p+1)/2$ , there exist arbitrarily large primes $q$ and $r$ such that the height $A(pqr) = h$.

Significance: This provides a constructive answer to the question of whether every natural number occurs as a height, specifically within the bounds allowed by a fixed $p$ . It serves as a major step toward "Conjecture 1" mentioned in the introduction.

B. Diameter Results (Corollaries 6, 7, and 8)

The paper explores the range of possible diameters $D(pqr)$ for a fixed $p$ :

Existence: For a fixed $p$ , the diameter $D(pqr)$ can assume at least $(p+1)/2$ different values.
Even Diameters: Every even number $\geq 6$ is a diameter of some $\Phi_{pqr}$ for a sufficiently large $p$ .
Odd Diameters: For every odd number $d \geq 3$ , there is at least one prime $p$ such that $D(pqr) = d$ is soluble.

C. Explicit Evaluation (Theorem 3)

The author provides a complete classification of the coefficient sets $A_T$ for triples satisfying specific modular conditions. This allows for the exact determination of $A(T)$ based on the values of $t$ and $s$ (the residues of $q$ and $r$ modulo $p$ ).

4. Significance

This work is significant for several reasons:

Resolving Long-standing Questions: It moves the field closer to a complete characterization of the possible heights of cyclotomic polynomials.
Bridging Algebra and Combinatorics: By treating cyclotomic polynomials as inclusion-exclusion polynomials, the author uses combinatorial counting to solve algebraic problems, providing a more robust framework for studying coefficients.
Precision: Unlike previous works that provided conditional results or bounds, this paper provides explicit constructions for $q$ and $r$ , proving that specific heights and diameters are not just possible, but can be achieved with arbitrarily large primes.