Theoretical study reveals explicit monodromy and height asymptotics in a polynomial family, demonstrating quantitative decay of exceptional rational specializations.
Key Points
To classify the rational reducibility locus, compute geometric and arithmetic monodromy, and determine the quantitative height distribution of exceptional parameters for the polynomial family f_{ℓ,t}(x) = x^{2ℓ} − x^ℓ − t.
Classified the rational reducibility locus and exceptional parameters using Kummer-theoretic methods and the Vahlen–Capelli criterion.
Identified the geometric covering as a Belyi-type map and computed geometric monodromy, arithmetic monodromy, branch cycles, and Galois closure genus.
Performed primitive lattice-point analysis to quantify the asymptotic density of exceptional rational parameters ordered by projective height B.
Determined the exact genus of the Galois closure as g(X_ℓ) = ((ℓ − 1)(ℓ − 2))/2 and characterized the geometric Galois group as a wreath-product-type structure.
Established the asymptotic exceptional count E_ℓ(B) = (3A/π²)B + O_ℓ(√B log B) with A = √5/2 + 2 log((1 + √5)/2), proving a relative density decay of E_ℓ(B)/N(B) = A/(4B) + O_ℓ(log B / B^(3/2)).
Proved explicit factorization dynamics and thin-set bounds for arithmetic specializations evaluated at Fibonacci number ratio convergents.