TY - JOUR

T1 - On a variant of the Ailon–Rudnick theorem in finite characteristic

AU - Ghioca, Dragos

AU - Hsia, Liang-Chung

AU - Tucker, Thomas J.

N1 - Funding Information:
The research of the first author was partially supported by an NSERC Discovery grant. The second author was supported by MOST grant 104-2115-M-003-004-MY2. The third author was partially supported by NSF Grant DMS-1501515.

PY - 2017/2/20

Y1 - 2017/2/20

N2 - Let L be a field of characteristic p, and let a, b, c, d ε L(T). Assume that a and b are algebraically independent over Fp. Then for each fixed positive integer n, we prove that there exist at most finitely many λ ε L satisfying f(a(λ)) = c(λ) and g(b(λ)) = d(λ) for some polynomials f, g ε Fpn[Z] such that f(a) ≠ c and g(b) ≠ d. Our result is a characteristic p variant of a related statement proven by Ailon and Rudnick.

AB - Let L be a field of characteristic p, and let a, b, c, d ε L(T). Assume that a and b are algebraically independent over Fp. Then for each fixed positive integer n, we prove that there exist at most finitely many λ ε L satisfying f(a(λ)) = c(λ) and g(b(λ)) = d(λ) for some polynomials f, g ε Fpn[Z] such that f(a) ≠ c and g(b) ≠ d. Our result is a characteristic p variant of a related statement proven by Ailon and Rudnick.

KW - Ailon-Rudnick theorem

KW - Weil height

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M3 - Article

AN - SCOPUS:85013821286

VL - 23

SP - 213

EP - 225

JO - New York Journal of Mathematics

JF - New York Journal of Mathematics

SN - 1076-9803

ER -