2004/01/26 by Joseph H. Silverman
Mathematics · #math.NT #math.AG #msc:11T55 #msc:11R58 #msc:11D61
published as New York Journal of Math. (electronic) 10 (2004), 37--43
arxiv created 2004/01/26 · arxiv updated 2009/12/01
Ailon and Rudnick have shown that if a,b ∈ C[T] are multiplicatively independent polynomials, then °(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 °(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., gec(n,p)=1).