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Short Salem polynomials

This paper provides a complete classification of Salem polynomials of length 5 and, assuming Lehmer's Conjecture, characterizes all but finitely many length-6 Salem polynomials by identifying 12 infinite families and listing 126 specific exceptions, alongside a comprehensive table of short polynomials for Salem numbers below the smallest Pisot number.

Original authors: James McKee, Chris Smyth

Published 2026-05-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: James McKee, Chris Smyth

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 detective trying to find the "shortest" possible secret codes that unlock a very specific type of mathematical lock. These locks are called Salem numbers.

In this paper, authors James McKee and Chris Smyth act as master code-breakers. They have spent their time hunting for these numbers, but with a very specific rule: they are only looking for codes (polynomials) that are "short."

What is a "Short" Code?

In the world of math, a polynomial is like a recipe made of numbers and variables (like $z$ ). The "length" of this recipe isn't how many ingredients it has, but the sum of the absolute sizes of its numbers.

Imagine a recipe: $z^{12} - z^7 - z^6 - z^5 + 1$ .
The numbers are $1, -1, -1, -1, 1$.
If you ignore the minus signs and add them up ( $1+1+1+1+1$ ), the length is 5.

The authors want to find every single "Salem number" that can be unlocked by a code with a length of 5 or 6.

The Mystery of Length 5: The "Perfect 17"

First, the authors tackled the shortest possible codes: Length 5.

The Rule: They proved that you can't make a Salem number with a code shorter than 5. It's like trying to build a house with only 3 bricks; it just won't stand up.
The Discovery: They found exactly 17 unique codes of length 5.
The Result: These 17 codes unlock 13 different Salem numbers. One of these numbers is famous: it's the smallest Salem number ever discovered (known as Lehmer's number). The authors confirmed that there are no other "hidden" Salem numbers with a code this short. They have a complete list, and the list is finished.

The Mystery of Length 6: The "Infinite Families" and the "Oddballs"

Next, they looked at codes of Length 6. This is where the plot thickens.

1. The Infinite Families (The Train Tracks)
They discovered that most Length-6 codes aren't random. They belong to 12 infinite families.

The Analogy: Imagine a train track. Once you find the pattern of the tracks, you know the train can go on forever. These 12 families are like 12 different train tracks that stretch on infinitely. You can generate an endless number of Salem numbers by following these specific patterns.

2. The Sporadic Oddballs (The Stray Cats)
But, not all Length-6 codes fit on these tracks. Some are "sporadic"—they are unique, one-off codes that don't follow the infinite patterns.

The Challenge: Because there are infinitely many numbers, how do you know you haven't missed a stray cat?
The Solution: The authors used powerful math tools to prove that there are only a finite number of these oddballs.
The List: They managed to find and list 126 of these "stray cat" codes.
- Note: They found 116 of them that correspond to Salem numbers smaller than a famous number called the "smallest Pisot number" (think of this as a specific speed limit on the number line).
- They found 10 more that are slightly larger.
- The Caveat: They say, "If a famous math guess called 'Lehmer's Conjecture' is true (which most people believe it is), then our list of 126 is 100% complete." If that guess is wrong, there might be a few more very tiny numbers hiding out there, but they are so small they are practically invisible.

The "Pisot" Connection

To understand how they found these infinite families, the authors used a helper called a Pisot number.

The Analogy: Think of a Pisot number as a "seed." If you plant this seed and grow a specific type of mathematical plant (using a method discovered by a mathematician named Salem), it grows into a Salem number.
The authors showed that for every "seed" (Pisot polynomial) of a certain type, you can grow an infinite line of Salem numbers. They cataloged exactly which seeds grow which lines.

The Final Treasure Map

The paper ends with a massive table (Table 4).

This is a "Who's Who" of all the known Salem numbers that are smaller than the smallest Pisot number.
For each number, they provide the "shortest code" (the polynomial) that unlocks it.
They also tell you the "degree" (how complex the code is) and the "shortness" (the sum of the numbers).

Summary

In simple terms, McKee and Smyth have:

Finished the job for the shortest codes (Length 5): There are exactly 17 of them.
Mapped the landscape for the next shortest codes (Length 6): They found 12 infinite patterns and a specific list of 126 unique exceptions.
Created a reference guide for all the small Salem numbers currently known to humanity.

They haven't found a new use for these numbers in the real world (like medicine or engineering); instead, they have simply solved a massive puzzle in pure mathematics, ensuring that for these specific "short" codes, the list is as complete as it can possibly be.

Technical Summary: Short Salem Polynomials

Problem Statement
The paper addresses the classification of Salem polynomials based on their "length," defined as the sum of the absolute values of their coefficients. A Salem number $\tau > 1$ is a real algebraic integer whose Galois conjugates (excluding $\tau$ ) have modulus at most 1, with at least one having modulus exactly 1. A Salem polynomial is the minimal polynomial of a Salem number, potentially multiplied by cyclotomic polynomials. The authors focus on polynomials of short length, specifically length 5 and length 6. While it is established that monic reciprocal integer polynomials of length $\le 4$ are cyclotomic, the minimal length for a Salem polynomial is 5. The paper seeks to provide a complete enumeration of Salem polynomials of length 5 and a structural classification of those of length 6.

Methodology
The authors employ a combination of analytic number theory, algebraic manipulation, and computational search algorithms:

Analytic Bounds and Interlacing: For length 5, the authors analyze the function $s(\theta)$ derived from the polynomial evaluated on the unit circle ( $z = e^{i\theta}$ ). They utilize Lemma 2.1 regarding the interlacing of zeros of $f(\theta)$ and $f'(\theta)$ to bound the number of solutions to $s(\theta)=1$ . This allows them to prove that for sufficiently large degrees, the polynomial cannot satisfy the conditions to be a Salem polynomial, thereby restricting the search to a finite set of parameters.
Salem's Construction and Pisot Sequences: For length 6, the paper relies on Salem's method of constructing sequences of Salem numbers converging to Pisot numbers. Specifically, for an irreducible Pisot polynomial $P(z)$ and $\epsilon \in \{-1, 1\}$ , the sequence $P_n(z) = z^n P(z) + \epsilon P^*(z)$ is analyzed. Proposition 2.5, largely based on Boyd's work, establishes conditions under which these polynomials possess a unique zero outside the unit circle (making them Salem or Pisot) and characterizes their cyclotomic factors.
Computational Search with Gap Bounds: To find sporadic (non-family) Salem polynomials of length 6, the authors adapt an algorithm from [10] to search for reciprocal polynomials within specific intervals $[a, b]$ . They utilize Gonçalves' inequality and Dobrowolski's lower bounds to constrain the search space. The search involves bounding the "gaps" between non-zero coefficients. When a gap cannot be bounded due to a zero of an intermediate polynomial in the interval, the authors identify this as a limit point corresponding to an infinite family or a specific sporadic case.
Cyclotomic Factor Analysis: The authors explicitly compute the cyclotomic factors of the infinite families using resultants and properties of roots of unity (Lemma 2.3), allowing them to determine the minimal polynomials and degrees of the resulting Salem numbers.

Key Contributions and Results

Classification of Length 5 (Theorem 1): The paper provides a complete list of all Salem polynomials of length 5. There are exactly 17 such polynomials associated with 13 distinct Salem numbers. These results confirm that no Salem numbers of shortness 5 exist below Lehmer's number ( $\lambda_0 \approx 1.176$ ), matching previous findings in [7] but providing a rigorous proof of completeness.
Classification of Length 6 (Theorem 2): The authors demonstrate that all but finitely many Salem polynomials of length 6 belong to one of 12 infinite families. These families are generated by Salem's construction using specific Pisot polynomials (trinomials and $z-2$ $z - 2$ ).
- The paper identifies 126 "sporadic" length-6 Salem polynomials with Salem numbers $\ge 1.01$ .
- Of these, 116 correspond to Salem numbers below the smallest Pisot number $\theta_0 \approx 1.3247$ , and 10 are above $\theta_0$ .
- The existence of these families and the finiteness of the sporadic set are proven, though the completeness of the sporadic list below 1.01 relies on the assumption that the search can be extended arbitrarily close to 1 (which is computationally impractical) or on the truth of Lehmer's Conjecture.
Cyclotomic Factors in Families (Theorem 3 & Proposition 5.1): The paper details the cyclotomic factors for the 12 infinite families. It provides explicit formulas for the minimal polynomials of the Salem numbers in these families, showing that for large $n$ , the degree of the minimal polynomial grows linearly with $n$ , minus a periodic correction term determined by the cyclotomic factors.
Comprehensive Tables: The authors provide:
- Table 1: The 17 length-5 Salem polynomials and their corresponding numbers.
- Table 2: The 126 known sporadic length-6 Salem polynomials (for numbers $\ge 1.01$ ).
- Table 4: A compilation of all known Salem numbers below the smallest Pisot number $\theta_0$ , listing their degrees, shortness, and minimal-degree short polynomials. This includes the sequence $\tau_n$ converging to $\theta_0$ and 53 sporadic numbers.

Significance
The paper claims to provide the first complete classification of Salem polynomials of length 5. For length 6, it establishes a structural dichotomy: the set of such polynomials consists of a finite number of sporadic examples and a countable set of infinite families. The authors note that their list of sporadic length-6 polynomials is complete for Salem numbers $\ge 1.01$ . They state that while extending this list to numbers arbitrarily close to 1 is impractical, the list would be complete if Lehmer's Conjecture (that $\lambda_0$ is the smallest Salem number) is true. The work refines the understanding of the distribution of Salem numbers near 1 and clarifies the relationship between Salem numbers, Pisot numbers, and cyclotomic factors through explicit polynomial constructions.