Factorisation in the ring of exponential polynomials

Abstract

We study factorisation in the ring of exponential polynomials and provide a proof of Ritt's factorisation theorem in modern notation and so gen-eralised as to deal with polynomial coefficients as well as with several variables. We do this in the more general context of a group ring of a divisible torsion-free ordered abelian group over a unique factorisation domain. We study factorisation in the group ring of a divisible torsion-free ordered abelian group over a unique factorisation domain. A natural instance of such a ring is the ring of exponential polynomials. Thus, in particular we give an account of Ritt's factorisation theorem for exponential polynomials. Ritt's paper [6] deals just with exponential polynomials with constant coefficients and his proof relies on some incongenial normalisations, which we show to be unnecessary. Hence our argument generalises more readily; in particular we remark that our proof applies to exponential polynomials in several variables. Even the case of polynomial coefficients does not seem to appear in the literature, other than for an allusion by Shields [10] to an unpublished manuscript of W. D. Bouwsma. The factorisation theorem for exponential polynomials is not just of intrinsic interest. It plays a critical role in the analysis of recurrence sequences that are divisibility sequences detailed in [1].