TY - JOUR

T1 - Sums of two squares in short intervals in polynomial rings over finite fields

AU - Bank, Efrat

AU - Bary-Soroker, Lior

AU - Fehm, Arno

N1 - Publisher Copyright:
© 2018 by Johns Hopkins University Press.

PY - 2018/8

Y1 - 2018/8

N2 - Landau’s theorem asserts that the asymptotic density of sums of two squares in the interval 1 ≤ n ≤ x is K/ √logx, where K is the Landau-Ramanujan constant. It is an old problem in number theory whether the asymptotic density remains the same in intervals |n−x| ≤ xϵ for a fixed ϵ and x → ∞. This work resolves a function field analogue of this problem, in the limit of a large finite field. More precisely, consider monic f0 ∈ Fq [T] of degree n and take ϵ with 1 > ϵ ≥ 2/n. Then the asymptotic density of polynomials f in the “interval” deg(f − f0) ≤ ϵn that are of the form f = A2 +TB2, A,B ∈ Fq [T] is 1/4n (2n/n) as q →∞. This density agrees with the asymptotic density of such monic f’s of degree n as q → ∞, as was shown by the second author, Smilanski, and Wolf. A key point in the proof is the calculation of the Galois group of f(−T2), where f is a polynomial of degree n with a few variable coefficients: The Galois group is the hyperoctahedral group of order 2nn!.

AB - Landau’s theorem asserts that the asymptotic density of sums of two squares in the interval 1 ≤ n ≤ x is K/ √logx, where K is the Landau-Ramanujan constant. It is an old problem in number theory whether the asymptotic density remains the same in intervals |n−x| ≤ xϵ for a fixed ϵ and x → ∞. This work resolves a function field analogue of this problem, in the limit of a large finite field. More precisely, consider monic f0 ∈ Fq [T] of degree n and take ϵ with 1 > ϵ ≥ 2/n. Then the asymptotic density of polynomials f in the “interval” deg(f − f0) ≤ ϵn that are of the form f = A2 +TB2, A,B ∈ Fq [T] is 1/4n (2n/n) as q →∞. This density agrees with the asymptotic density of such monic f’s of degree n as q → ∞, as was shown by the second author, Smilanski, and Wolf. A key point in the proof is the calculation of the Galois group of f(−T2), where f is a polynomial of degree n with a few variable coefficients: The Galois group is the hyperoctahedral group of order 2nn!.

UR - http://www.scopus.com/inward/record.url?scp=85050091763&partnerID=8YFLogxK

U2 - 10.1353/ajm.2018.0025

DO - 10.1353/ajm.2018.0025

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AN - SCOPUS:85050091763

VL - 140

SP - 1113

EP - 1131

JO - American Journal of Mathematics

JF - American Journal of Mathematics

SN - 0002-9327

IS - 4

ER -