On subword decomposition and balanced polynomials

Yossi Moshe

doi:10.1016/j.jnt.2006.07.007

On subword decomposition and balanced polynomials

Yossi Moshe

Research output: Contribution to journal › Article › peer-review

Abstract

Let H (x) be a monic polynomial over a finite field F = GF (q). Denote by N_a (n) the number of coefficients in Hⁿ which are equal to an element a ∈ F, and by G the set of elements a ∈ F^× such that N_a (n) > 0 for some n. We study the relationship between the numbers (N_a (n))_{a ∈ G} and the patterns in the base q representation of n. This enables us to prove that for "most" n's we have N_a (n) ≈ N_b (n), a, b ∈ G. Considering the case H = x + 1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind.

Original language	English
Pages (from-to)	224-240
Number of pages	17
Journal	Journal of Number Theory
Volume	123
Issue number	1
DOIs	https://doi.org/10.1016/j.jnt.2006.07.007
State	Published - 1 Jan 2007
Externally published	Yes

Keywords

Asymptotic frequency
Linear cellular automata
Pascal's triangle
Stirling numbers

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jnt.2006.07.007

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N2 - Let H (x) be a monic polynomial over a finite field F = GF (q). Denote by Na (n) the number of coefficients in Hn which are equal to an element a ∈ F, and by G the set of elements a ∈ F× such that Na (n) > 0 for some n. We study the relationship between the numbers (Na (n))a ∈ G and the patterns in the base q representation of n. This enables us to prove that for "most" n's we have Na (n) ≈ Nb (n), a, b ∈ G. Considering the case H = x + 1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind.

AB - Let H (x) be a monic polynomial over a finite field F = GF (q). Denote by Na (n) the number of coefficients in Hn which are equal to an element a ∈ F, and by G the set of elements a ∈ F× such that Na (n) > 0 for some n. We study the relationship between the numbers (Na (n))a ∈ G and the patterns in the base q representation of n. This enables us to prove that for "most" n's we have Na (n) ≈ Nb (n), a, b ∈ G. Considering the case H = x + 1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind.

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KW - Linear cellular automata

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KW - Stirling numbers

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On subword decomposition and balanced polynomials

Abstract

Keywords

ASJC Scopus subject areas

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