New York Journal of Mathematics Volume 10 (2004) 37-43
 Volume 10

Joseph H. Silverman

Common divisors of an-1 and bn-1 over function fields

 Published: January 20, 2004 Keywords: greatest common divisor, function field Subject: 11T55; 11R58; 11D61

Abstract
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\notdivide n).

Author information

Mathematics Department, Box 1917, Brown University, Providence, RI 02912 USA
jhs@math.brown.edu