On a problem of Sidon for polynomials over finite fields

Wentang Kuo ; Shuntaro Yamagishi

Acta Arithmetica, Tome 172 (2016), p. 239-254 / Harvested from The Polish Digital Mathematics Library

Résumé

Let ω be a sequence of positive integers. Given a positive integer n, we define rₙ(ω) = |(a,b) ∈ ℕ × ℕ : a,b ∈ ω, a+b = n, 0 < a < b|. S. Sidon conjectured that there exists a sequence ω such that rₙ(ω) > 0 for all n sufficiently large and, for all ϵ > 0, $lim_{n→ ∞} rₙ(ω)/n^{ϵ} = 0$. P. Erdős proved this conjecture by showing the existence of a sequence ω of positive integers such that log n ≪ rₙ(ω) ≪ log n. In this paper, we prove an analogue of this conjecture in $_{q}[T]$, where $_{q}$ is a finite field of q elements. More precisely, let ω be a sequence in $_{q}[T]$. Given a polynomial $h ∈ _{q}[T]$, we define $r_{h}(ω) = |{(f,g) ∈ _{q}[T] × _{q}[T]: f,g ∈ ω, f + g = h, deg f, deg g ≤ deg h, f ≠ g}|$. We show that there exists a sequence ω of polynomials in $_{q}[T]$ such that $deg h ≪ r_{h}(ω) ≪ deg h$ for deg h tending to infinity.

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Publié le : 2016-01-01

Zbl 06622302

EUDML-ID : urn:eudml:doc:286614