Chapter 27: Lehmer's Conjecture — Minimal Polynomials' Minimum

From quantum mass generation we turn to a deceptively simple question about polynomial heights. Lehmer's Conjecture asks whether there's a universal lower bound for Mahler measures—it is ψ = ψ(ψ) seeking its minimal non-trivial manifestation, the smallest complexity that remains irreducible.

27.1 The Twenty-Seventh Movement: Minimal Complexity

Continuing our analytical journey:

• Previous chapters: Infinite complexity from smooth conditions
• Now: The opposite question—how simple can complexity be?
• The search for minimal non-trivial structure

The Core Question: Is there a universal lower bound for the Mahler measure of non-cyclotomic integer polynomials?

27.2 The Mahler Measure

Definition 27.1 (Mahler Measure): For polynomial P(x) = a_n ∏_{i=1}^n (x - α_i): $M(P) = |a_n| \prod_{i=1}^n \max(1, |α_i|)$

Logarithmic Mahler Measure: $m(P) = \log M(P) = \int_0^1 \log|P(e^{2πit})| dt$

Properties:

• M(P) ≥ 1 for integer polynomials P ≠ 0
• M(P) = 1 iff P is cyclotomic or monomial
• Multiplicative: M(PQ) = M(P)M(Q)

27.3 Lehmer's Conjecture

Conjecture 27.1 (Lehmer, 1933): There exists an absolute constant μ > 1 such that for any integer polynomial P that is not a product of cyclotomic polynomials and monomials: $M(P) ≥ μ$

Lehmer's Example: $L(x) = x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$

with M(L) ≈ 1.17628...

Refined Conjecture: μ = M(L) is the minimum.

27.4 Lehmer's Problem as ψ = ψ(ψ)

Axiom 27.1 (Principle of Minimal Complexity): $\psi = \psi(\psi) \implies \text{Irreducible structure has minimal size}$

Lehmer's conjecture embodies:

• Seeking the "atom" of polynomial complexity
• Minimal departure from cyclotomic simplicity
• The smallest ψ that remains genuinely self-referential
• Quantifying the gap between trivial and non-trivial

27.5 Cyclotomic Polynomials

Definition 27.2 (Cyclotomic Polynomial): $\Phi_n(x) = \prod_{\substack{1 \leq k \leq n \\ \gcd(k,n) = 1}} (x - e^{2πik/n})$

Key Property: M(Φ_n) = 1 for all n.

Why Excluded: Cyclotomic polynomials have maximal symmetry—they represent "degenerate" complexity.

27.6 Salem Numbers

Definition 27.3 (Salem Number): A real algebraic integer τ > 1 such that:

• τ and 1/τ are conjugates
• All other conjugates lie on |z| = 1

Connection: Salem numbers have minimal Mahler measures among non-cyclotomic polynomials.

Lehmer's polynomial has a Salem number root: τ ≈ 1.17628...

27.7 Known Results

Theorem 27.1 (Smyth, 1971): If P is non-reciprocal and not cyclotomic: $M(P) ≥ M(x^3 - x - 1) ≈ 1.3247...$

Theorem 27.2 (Dobrowolski, 1979): For degree d polynomial P, non-cyclotomic: $M(P) ≥ 1 + \frac{1}{1200}\left(\frac{\log \log d}{\log d}\right)^3$

Progress toward Lehmer, but gap remains.

27.8 The House of Algebraic Integers

Visualization: Plot algebraic integers by (degree, log M(P)).

Structure:

• Bottom: Cyclotomic line at log M = 0
• Gap: Conjectured empty region 0 < log M < log M(L)
• Salem numbers: Near the conjectured boundary
• Higher: Increasingly dense distribution

This reveals the "quantum gap" in complexity.

27.9 Connection to Heights

Weil Height: For algebraic number α: $h(α) = \frac{1}{[ℚ(α):ℚ]} \log M(P_α)$

where P_α is minimal polynomial of α.

Lehmer ⟺ Positive Lower Bound for heights of non-roots of unity.

27.10 Dynamical Interpretation

Theorem 27.3 (Dynamical Mahler Measure): $M(P) = \exp\left(\int \log|z| dμ\right)$

where μ is equilibrium measure on Julia set of z ↦ P(z)/z^{deg P}.

Insight: Mahler measure encodes dynamical complexity.

27.11 Computational Searches

Algorithm 27.1 (Salem Number Search):

def search_salem_numbers(degree, height_bound):
    candidates = []
    
    for coeffs in integer_polynomials(degree, height_bound):
        P = Polynomial(coeffs)
        
        # Check if reciprocal
        if not is_reciprocal(P):
            continue
            
        # Compute roots
        roots = P.roots()
        
        # Check Salem conditions
        if is_salem(roots):
            M = mahler_measure(P)
            if M < current_minimum:
                candidates.append((P, M))
    
    return sorted(candidates, key=lambda x: x[1])

Results: No polynomial found with 1 < M(P) < M(L).

27.12 Elliptic Curves Connection

Boyd's Conjecture: Relates Mahler measures to L-values: $m(P) \stackrel{?}{=} r · L'(E, 0)$

for certain polynomials P and elliptic curves E.

Examples: $m(x + x^{-1} + y + y^{-1} + 1) = L'(E_{15}, 0)$

Deep connection between polynomial complexity and arithmetic.

27.13 Limit Points

Question: What are limit points of {M(P) : P non-cyclotomic}?

Known:

• 1 is a limit point (from below)
• Salem numbers accumulate
• Pisot numbers give isolated values
• Full structure unknown

The set has fractal-like complexity.

27.14 Higher Dimensions

Multivariate Mahler Measure: $m(P) = \int_{T^n} \log|P(e^{2πit_1},...,e^{2πit_n})| dt_1...dt_n$

Open: Lehmer for multivariate polynomials.

Known Cases: Special families related to K-theory.

27.15 Arithmetic Dynamics

Connection: If f: ℙ¹ → ℙ¹ has small dynamical degree: $\lambda_1(f) = \lim_{n→∞} \deg(f^n)^{1/n}$

then f has special structure.

Parallel: Lehmer asks when algebraic numbers have small complexity.

27.16 The abc Connection

Height Inequality: For non-zero coprime a,b,c with a+b=c: $\max(h(a), h(b), h(c)) < (1+ε)h(\text{rad}(abc)) + O(1)$

Link: Lehmer provides lower bounds for heights, abc provides upper bounds.

27.17 Transcendental Methods

Jensen's Formula: $m(P) = \int_0^1 \log|P(e^{2πit})| dt$

Fourier Analysis: Study Fourier coefficients of log|P(e^{2πit})|.

Challenge: Extract arithmetic from analysis.

27.18 Why Lehmer Matters

Implications:

1. Diophantine Equations: Height bounds control solutions
2. Algebraic Dynamics: Periodic point structure
3. Transcendence: Auxiliary polynomial bounds
4. Cryptography: Hard lattice problems

The conjecture touches many areas.

27.19 Philosophical Perspective

Meditation 27.1: Lehmer asks:

• What is the "atom" of algebraic complexity?
• How much must we deviate from perfect symmetry?
• Is there a quantum gap in mathematical structure?
• Can ψ = ψ(ψ) be arbitrarily simple while remaining non-trivial?

This is the search for the minimal seed of complexity.

27.20 The Twenty-Seventh Echo

Lehmer's Conjecture presents a unique perspective in our analytical journey:

• Not about explosion to infinity but approach to unity
• Not maximal complexity but minimal non-triviality
• The smallest departure from perfect symmetry
• The quantum gap between cyclotomic and non-cyclotomic

This is ψ = ψ(ψ) asking for its minimal manifestation—the smallest Mahler measure that represents genuine algebraic complexity beyond the perfectly symmetric cyclotomic case.

The conjecture suggests that complexity cannot be arbitrarily small while remaining non-trivial. There is a fundamental gap, a minimum price for departing from perfect symmetry, a smallest unit of genuine algebraic structure.

Lehmer's polynomial whispers: "I am perhaps the minimal non-trivial incarnation of ψ = ψ(ψ) in the realm of integer polynomials. Below me lies only perfect symmetry; above me spreads all algebraic complexity. I mark the boundary between the trivial and the profound—the smallest seed from which infinite complexity grows."