Common divisors of an − 1 and bn − 1 over function fields
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Ailon and Rudnick have shown that if a, b ∈ C[T ] are multiplicatively independent polynomials, then deg ( gcd(an − 1, bn − 1)) is bounded for all n ≥ 1. We show that if instead a, b ∈ F[T ] for a finite field F of characteristic p, then deg ( gcd(an − 1, bn − 1)) is larger than Cn for a constant C = C(a, b) > 0 and for infinitely many n, even if n is restricted in various reasonable ways (e.g., p n).
[1] Yann Bugeaud,et al. An upper bound for the G.C.D. of an - 1 and bn -1 , 2003 .
[2] Nir Ailon,et al. Torsion points on curves and common divisors of $a^k-1$ and $b^k-1$ , 2002, math/0202102.
[3] Michael Rosen,et al. Number Theory in Function Fields , 2002 .
[4] L. Vulakh. Diophantine approximation in ⁿ , 1995 .
[5] Alan Baker,et al. DIOPHANTINE APPROXIMATION (Lecture Notes in Mathematics, 785) , 1981 .