On the integrality of nth roots of generating functions

Abstract

Motivated by the discovery that the eighth root of the theta series of the E_8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f∈R (where R=1+xZ〚x〛) can be written as f=g^n for g∈R, n⩾2. Let P_n:={g^n|g∈R} and let μ_n:=n∏_p|_np. We show among other things that (i) for f∈R, f∈P_n⇔f(mod μ_n)∈P_n, and (ii) if f∈P_n, there is a unique g∈P_n with coefficients mod μ_n/n such that f≡g^n(mod μ_n). In particular, if f≡1(mod μ_n) then f∈P_n. The latter assertion implies that the theta series of any extremal even unimodular lattice in R^n (e.g. E_8 in R^8) is in P_n if n is of the form 2^i3^j5^k (i⩾3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order Reed–Muller code of length 2^m is in P_2r(and similarly that the theta series of the Barnes–Wall lattice BW_2m is in P_2m). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f∈P_n (n⩾2) with coefficients restricted to the set {1,2,…,n}.