 

Joseph H. Silverman
Common divisors of a^{n}1 and b^{n}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(a^{n}1,b^{n}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(a^{n}1,b^{n}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

