(Prior to publication, this paper should be cited as arXiv:0710.1902.)

In the 1920's, Ritt studied the operation of functional composition  g(x) o h(x) = g(h(x))  on complex rational functions. In the case of polynomials, he described all the ways in which a polynomial can have multiple `prime factorizations' with respect to this operation. Despite significant effort by Ritt and others, little progress has been made towards solving the analogous problem for rational functions. In this paper we prove analogues of Ritt's polynomial decomposition results for decompositions of Laurent polynomials, namely, rational functions whose denominator is a monomial.

Ritt's polynomial decomposition results have been applied in various wide-ranging contexts, including:

• The Bilu–Tichy classification of all  f, g  in  Z[X]  for which the Diophantine equation  f(x) = g(y) has infinitely many integer solutions;
• The Medvedev–Scanlon classification of affine varieties invariant under a coordinatewise polynomial action;
• Pakovich's classification of  f, g  in C[X] having identical preimages on some infinite compact subsets of  C;
• The Ghioca–Tucker–Zieve proof that nonlinear complex polynomials have orbits with infinite intersection if and only if they have a common iterate;
• Zannier's proof that, if  f  is a formal Laurent series over  C  which is not a rational function, then the set of positive integers  m  for which  f(X^m) ∈ C(X)[f] consists of the powers of a single integer.

Our analogue of the "first theorem of Ritt" is as follows. If a Laurent polynomial  f  is written in two ways as the composition of indecomposable rational functions,  f = p₁ o p₂ o ... o p_r = q₁ o q₂ o ... o q_s,  then the sequences  (deg(p₁), ..., deg(p_r))  and  (deg(q₁), ..., deg(q_s))  are permutations of one another (so  r = s). Moreover, there is a finite sequence of decompositions of  f  into indecomposables, which contains both  p₁ o p₂ o ... o p_r  and  q₁ o q₂ o ... o q_s , such that consecutive decompositions in the sequence differ only in that two adjacent indecomposables in the first decomposition are replaced in the second by two others having the same composition.

The "second theorem of Ritt", together with some auxiliary results proved by Ritt, enables one to solve the equation  a o b  = c o d  in polynomials  a, b, c, d.  These auxiliary results do not generalize to Laurent polynomials, so the solution to this equation in Laurent polynomials is more difficult. Our main result is a precise description of all rational functions  a, b, c, d  such that  a o b  = c o d  is a Laurent polynomial.

We make crucial use of results of Bilu–Tichy and Avanzi–Zannier, which determine the curves of certain forms having a genus-zero factor with at most two points at infinity. These results reduce our problem to the determination of all parametrizations of these curves, and to the study of decompositions of certain explicit families of Laurent polynomials. We solve these two subproblems via several types of arguments.

Additional comment added October 2007:  Alternate proofs of some of our results were discovered by Pakovich.

Additional comment added December 2007:  Simpler proofs of the Laurent polynomial analogue of Ritt's first theorem have been discovered by Kuperberg, Lyons and Zieve and Muzychuk and Pakovich. However, these arguments do not yield any simplification to the proof of the more difficult Laurent polynomial analogue of Ritt's second theorem.

Additional comment added May 2008:  The results of this paper have been used by Watt to compute functional decompositions of symbolic polynomials, that is, polynomials whose exponents are themselves integer-valued polynomials.

Michael Zieve:  home page   publication list