Jon,

       I read through your thesis this afternoon and while I 
   don't know this area at all well (and hence cannot vouch 
   for how much of this would be new and publishable),
   it seems like a very impressive piece of work!  

   I have a few comments for you and Rafe:

   General comment:  The Chapter 1 (Introduction) seems *very 
   terse* to me.  I guess how you react to what I'm going to say 
   depends on the intended audience you are writing for.  If 
   you are thinking of this as the manuscript for a journal
   article that you are going to submit, then you could get away
   with doing less.  But if you are thinking of this as the 
   background chapter and literature review of a typical thesis, 
   you would want to do a lot more.  For instance, 

     *  Doing some more background examples would be nice (but
        not necessary for a projected journal submission).

     *  What are the results from [8], Section 4.1 (cited on 
        page 1) that imply "we don't need to worry" except for 
        for a few quadratic polynomials?   

     *  For me, the material from subsection 2.1 on the definition
        of the \Phi_k could have come earlier (in the Introduction), 
        because you really need to have some of that in mind to 
        understand your results.  And depending on the intended audience,
        you might want to explain exactly how the Moebius function
        works(!)  

     *  The statements of Theorems 1.1 and 1.2 on page 2 seemed
        very cryptic at first to this (nonexpert) reader.  Given just
        the definition of the \Phi_k to work with, in the statement of
        Theorem 1.1, it is not clear at all that there are any
        f(x) and infinite families of $k$ such that \Phi_k(x) has 
        a number of irreducible factors bounded above by \ell 
        independent of k. So I think it would be good to split 
        up those statements and reorganize them somewhat:

           - take the irreducibility statement from Theorem 1.2
             and state that first, 

           - then do Theorem 1.1, 

           - then add a Corollary giving the ``In particular''
             part of Theorem 1.2 for the polynomials and k for
             which you can show irreducibility of \Phi_k.

        Plus, put in more explanation of how everything fits together.
        It was my impression that the family of examples from Theorem 1.2
        is new, extending the Lemma 4.1 from [5].  It would be good to make
        that clearer if in fact that is the case(!)

   Specific comments:

   1)  The first paragraph of the introduction was somewhat rough.
       Maybe I'm being pedantic, but shouldn't the first sentence
       be ``maps defined by polynomials in the ring ${\mathbb Z}[x]$.''
       Then I would interchange sentences 3 and 4 (put the discussion
       of the minimal period first).  Then, the terminology "$k$-cycle"
       bothered me.  It sounded to me as though the ``cycle'' should
       be the orbit under the interation of $f(x)$, not the particular
       representative of the orbit.  If this is standard terminology,
       I guess it's OK, but it has the potential to create confusion.

   2)  The notation for wreath products is introduced without any
       prior warning on line 6 of page 2.  Since this is only one of 
       several different notations for this construction, it would
       be good at least to say what the ``wr'' notation means
       at this point.  

   3)  Picky typesetting comment:  last line of page 3 -- really should
       have a bit of space between the $\deg$ and the $f$

   4)  Page 4, line 18 -- why the subscript $f$ in $\Phi_{m,f}(x)$.
       Isn't the $f(x)$ understood?

   5)  Page 5, line 6 -- there's a typo in the polynomial.  The 
       $3x^7$ term should be $3x^3$.  Also, the discussion of this
       example is confusing for the reader because you stop 
       two times to state general results and discuss them, 
       then continue with the example.  Give a few more signposts to 
       help the reader follow exactly where you are.  For instance, 
       you might put in a reference number (``Example 2.x.'') and 
       after the Theorem 2.7 and Corollary 2.8, say ``Now we return to 
       Example 2.x.''

   6)  Pages 9 and 10 -- most people, books, etc. say ``inclusion-exclusion 
       principle'' or ``principle of inclusion and exclusion.''  ``Principle
       of inclusion exclusion'' sounds a bit odd.

   7)  Picky typesetting comment:  In the proof of Theorem 3.3 on page 12,
       it looks strange to have the periods inside the parentheses for
       different parts of the statement.  You didn't have periods in
       the statement, so don't put them in now.

   8)  Page 14, last line: The first wreath product on this line
       wr has a capital ``W''

   9)  Page 15 line 3 -- the middle part of this display doesn't
       really add anything.  I would delete it and just keep
       the first and third parts.

  10)  Page 16, line 4 -- ``Every'' should be ``every''

  11)  Page 18, line 6 -- Shouldn't $\Phi_{q^s}(0)$ be $\Phi_{p^s}(0)$?

  12)  Page 18, proof of Theorem 4.2 -- using $k$ as the exponent on 
       $p$ in several places here is slightly confusing because you
       also have the $k$ for the length of the cycle.  Try to find a 
       different letter for the exponents.  Also what is the double
       vertical bar notation at the end of the first line?  You have
       not used that before.

  13)  Page 18, lines -10 and -8 --  When you set $x = f^{p^{s-1}}$, I
       think you mean $f^{p^{s-1}}(0)$, and then there is a missing
       $(0)$ on line -8 in the second order term of the expansion.

  14)  Page 19, line -7 -- The (0) at the end of the last term should 
       be ``inside'' the square.  
        
  Hope this is useful, and let me know if you have any questions about
  my suggestions.

                                           John Little