Rational Specializations, Belyi Monodromy, and Exceptional Height Counts for x2` − x` − t

Abstract

This preprint studies rational specialization, monodromy, and arithmetic statistics for the one-parameter polynomial family fℓ,t(x) = x^(2ℓ) − x^ℓ − t, ℓ ≥ 2. The family is analyzed simultaneously as a polynomial over ℚ(t), as a branched cover of the projective line, and as a source of arithmetic specializations over ℚ. The rational reducibility locus is classified completely using binomial irreducibility and Kummer-theoretic methods. The exceptional specializations are described by explicit parametrized families, including all prime-power and quartic obstructions arising from the Vahlen–Capelli criterion. Outside these loci, the specialized polynomial retains the expected irreducibility behavior. Geometrically, the underlying map is identified as a Belyi-type covering with explicitly determined branch locus and ramification partitions. The geometric and arithmetic monodromy groups are computed, together with the constant field, branch-cycle structure, center, commutator subgroup, and abelianization. The geometric Galois group is a wreath-product-type group, while the arithmetic group acquires the additional cyclotomic multiplier action. The genus of the Galois closure is determined exactly: g(Xℓ) = ((ℓ − 1)(ℓ − 2))/2. The arithmetic specialization problem is then studied by projective height. The exceptional rational parameters are counted explicitly rather than merely shown to form a thin set. Primitive lattice-point analysis yields the asymptotic Eℓ(B) = (3A/π²)B + Oℓ(√B log B), where A = √5/2 + 2 log((1 + √5)/2). Relative to the total number N(B) of rational parameters of height at most B, this gives Eℓ(B)/N(B) = A/(4B) + Oℓ(log B / B^(3/2)). Thus the exceptional specializations have density tending to zero with an explicit leading constant and quantitative decay rate. The paper also develops a distinguished arithmetic specialization connected with Fibonacci numbers, proves the corresponding factorization and perfect-power exceptional behavior, and studies the associated Möbius dynamics and asymptotic structure of powered roots. The resulting framework combines polynomial specialization, Galois theory, Kummer theory, Belyi maps, monodromy, Diophantine analysis, thin sets, and height counting in a single explicit family. (Archived prior working draft description) This paper applies Hilbert’s irreducibility theorem to the two-variable polynomial family f(x,t) = x^{2ℓ} - x^ℓ - t over Q. It is proved that f(x,t) is irreducible over Q(t), and that for almost all rational values t₀ ∈ Q, the specialized polynomial f(x,t₀) remains irreducible over Q. The family includes as a special case the golden-ratio polynomial P_ℓ(x) = x^{2ℓ} - x^ℓ - 1, whose splitting field is Q(φ^{1/ℓ}, ζ_ℓ), where φ = (1+√5)/2 is the golden ratio. The paper connects Hilbert’s theorem with continued-fraction approximants of φ, showing that for almost all rational convergents tᵢ = F_{i+1}/F_i, the specialization x^{2ℓ} - x^ℓ - tᵢ is irreducible over Q. Quantitative bounds are provided for the thin exceptional set and for convergence of powered continued-fraction sequences.