On a problem of Sidon for polynomials over finite fields

• Autorzy: Wentang Kuo , Shuntaro Yamagishi
• Języki publikacji: EN

Abstrakty

EN

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 $𝔽_{q}[T]$, where $𝔽_{q}$ is a finite field of q elements. More precisely, let ω be a sequence in $𝔽_{q}[T]$. Given a polynomial $h ∈ 𝔽_{q}[T]$, we define
$r_{h}(ω) = |{(f,g) ∈ 𝔽_{q}[T] × 𝔽_{q}[T]: f,g ∈ ω, f + g = h, deg f, deg g ≤ deg h, f ≠ g}|$.
We show that there exists a sequence ω of polynomials in $𝔽_{q}[T]$ such that
$deg h ≪ r_{h}(ω) ≪ deg h$
for deg h tending to infinity.

Kategorie tematyczne

• 11T55: Arithmetic theory of polynomial rings over finite fields
• 11B83: Special sequences and polynomials
• 11K31: Special sequences

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2016
• Tom: 174
• Numer: 3
• Strony: 239-254

Daty

wydano

2016

Twórcy

autor

Wentang Kuo

• Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

autor

Shuntaro Yamagishi

• Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

Identyfikatory

DOI

10.4064/aa8252-3-2016