Arithmetic Properties of Generalized Rikuna Polynomials

Chonoles, Z.; Cullinan, John; Hausman, H.; Pacelli, Allison M.; Pegado, S.; Wei, Fu‐Tsun

doi:10.5802/pmb.2

Cited by 3 publications

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“…For the remainder of the paper, we fix K = Q(ζ + ). We derive two expressions for r n (x, t; ℓ) that do not appear in [3,17]. The first is a generalization of [15,Corollary 2.6].…”

Section: Preliminariesmentioning

confidence: 99%

“…By Equation (10), ν 3 (a 2,0 ) ≥ 5. The lower convex hull of the points {(1, 6), (3,4), (9, 0)} is marked by a dashed line.…”

Section: Shanks' Specialization: ℓ =mentioning

confidence: 99%

“…In a recent paper, Chonoles, Cullinan, Hausman, Pacelli, Pegado, and Wei [3] define a new family of postcritically finite maps, the generalized Rikuna polynomials, which also give rise to extensions with small Galois groups. These maps are named after a family of maps studied by Rikuna [15], which are defined as follows.…”

Section: Introductionmentioning

confidence: 99%

“…The possible (x − 1)-Newton polygons for r 2 (x, t) when t ≡ 1 (mod 3) are shown inFigure 6. By Proposition 4.2, the polygon is the lower convex hull of the points {(0, ν 3 (a 2,0 )),(1,6),(3,4), (9, 0)}. By Equation (10), ν 3 (a 2,0 ) ≥ 5.…”

mentioning

confidence: 97%

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Let ℓ > 2 be a positive integer, ζ ℓ a primitive ℓ-th root of unity, and K a number field containing ζ ℓ + ζ −1 ℓ but not ζ ℓ . In a recent paper, Chonoles et. al. study iterated towers of number fields over K generated by the generalized Rikuna polynomial, rn(x, t; ℓ) ∈ K(t) [x]. They note that when K = Q, t ∈ {0, 1}, and ℓ = 3, the only ramified prime in the resulting tower is 3, and they ask under what conditions is the number of ramified primes small. In this paper, we apply a theorem of Guàrdia, Montes, and Nart to derive a formula for the discriminant of Q(θ) where θ is a root of rn(x, t; 3), answering the question of Chonoles et. al. in the case K = Q, ℓ = 3, and t ∈ Z. In the latter half of the paper, we identify some cases where the dynamics of rn(x, t; ℓ) over finite fields yields an explicit description of the decomposition of primes in these iterated extensions.Date: July 12, 2018.

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“…For the remainder of the paper, we fix K = Q(ζ + ). We derive two expressions for r n (x, t; ℓ) that do not appear in [3,17]. The first is a generalization of [15,Corollary 2.6].…”

Section: Preliminariesmentioning

confidence: 99%

“…By Equation (10), ν 3 (a 2,0 ) ≥ 5. The lower convex hull of the points {(1, 6), (3,4), (9, 0)} is marked by a dashed line.…”

Section: Shanks' Specialization: ℓ =mentioning

confidence: 99%