Ailon and Rudnick have shown that if are multiplicatively independent polynomials, then is bounded for all . We show that if instead for a finite field of characteristic , then is larger than for a constant and for infinitely many , even if is restricted in various reasonable ways (e.g., ).
Let and be positive integers that are multiplicatively independent in and let . Bugeaud, Corvaja, and Zannier  recently showed that there is an so that In other words, and cannot share a common factor of significant size. Although elementary to state, the proof requires deep tools from Diophantine analysis, specifically Schmidt's subspace theorem .
Ailon and Rudnick  consider the analogous problem in which and are taken to be polynomials in . They prove the stronger result It is natural to consider the situation when and are polynomials in , where is a finite field of characteristic . In this case, some restriction on is certainly needed, since trivially In this paper we will show that for , even much stronger restrictions on the allowable values of do not allow one to prove an estimate analogous to (1), much less one as strong as (2).
Acknowledgements. The author would like to thank Gary Walsh for rekindling his interest in arithmetic properties of divisibility sequences and for drawing his attention to the papers  and , Felipe Voloch for a helpful discussion of Diophantine approximation in characteristic , and Andrew Granville and the referee for several suggestions that greatly improved this article.
As noted in the introduction, Ailon and Rudnick  prove that if are nonconstant polynomials that are multiplicatively independent in , then Suppose instead that , where is a finite field of characteristic . It is natural to ask if the Ailon-Rudnick estimate holds, at least if we require that . As the following example shows, the answer is no.
Before stating and proving a generalization of the example described in Section 1, we briefly recall some basic arithmetic facts about the rational function field . We start with some notation:
Proof. See [3, Theorem 2.2] for a proof. To see why this is the analogue of the classical prime number theorem, notice that there are monic polynomials of degree in , so the fact that is asymptotic to is analogous to the fact that is asymptotic to .
Proof. Let denote the power residue symbol ([3, Chapter 3]), and let be coset representatives for the elements with the property that Then one version [3, Proposition 3.6] of the -power reciprocity law in says that for any monic irreducible polynomial , (Note that our assumption that is odd ensures that either is even or else has characteristic 2.) In particular, the implication (4) is an immediate consequence of the fact that for every modulus .
Example 1 shows that for particular polynomials and , the polynomial can be large when . We first generalize this example to arbitrary polynomials and . We then consider more general exponent values and show that it is unlikely that there is any infinite “natural” set of exponents with the property that is finite for every .
Proof. To illustrate the main ideas, we start with the special case and , so as in Example 1, we look at exponents satisfying . More precisely, we will take For all , the group has order , so as long as , it follows that Hence for all sufficiently large , e.g., , we have and hence The “Prime Number Theorem for Polynomials” (Theorem 1) says that so we find that This completes the proof of the theorem for .
In order to obtain more general exponents, we take to have the form for a suitable choice of and . Then is divisible by primes for which both and are powers modulo . In order to exploit this weaker condition, we will use the function field versions of the power reciprocity law and Dirichlet's theorem on primes in arithmetic progression.
For now, we assume that , since this is the most interesting case. At the end of the proof we briefly indicate what to do if is divisible by the characteristic. Let be the smallest odd integer satisfying and let where is the usual Euler phi function. We note that (If we were aiming for better constants, we could take to be any power with and .)
For each power of , we let and we observe that since , we have It remains to show that for all .
Let be the (monic) least common multiple of and . We want to use Theorem 3 to find primes for which and are powers, but in order to apply Theorem 3, we need to work with a sufficiently large base field. More precisely, we work in , since our choice of ensures that the condition in Theorem 3 is satisfied.
We consider polynomials , i.e., monic irreducible polynomials of degree in satisfying . Then This proves that is divisible by every polynomial in , so we obtain the lower bound where recall that Finally, using the definition , we see that where the big- constant is independent of . This gives the desired result, with an explicit value for .
We are left to deal with the case that is divisible by the characteristic of . Write with . From the result proven above, we know that For these values of , it follows and that Thus the values with satisfying (5) show that Theorem 4 is true when .