On squares, cubes, ... in Z[[x]]*
Nadia Heninger, Eric M. Rains, Neil J.A. Sloane · arXiv (Cornell University) · 2005
Motivated by the discovery that the eighth root of the theta series of the E8 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 ∈ Z[[x]] ∗ can be written as f = g n for g ∈ Z[[x]] ∗ , n ≥ 2. Let Pn: = {g n | g ∈ Z[[x]] ∗ } and let µn: = n ∏ p|n p. We show among other things that (i) for f ∈ Z[[x]] ∗ , f ∈ Pn ⇔ f (mod µn) ∈ Pn, and (ii) if f ∈ Pn, up to sign there is a unique g ∈ Pn with coefficients mod µn/n such that f ≡ g n (mod µn). In particular, if f ≡ 1 (mod µn) then f ∈ Pn. The latter assertion implies that the theta series of any extremal even unimodular lattice in R n (e.g. E8 ∈ R 8) is in Pn if n is of the form 2 i 3 j 5 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 2m is in P2r (and similarly that the theta series of the Barnes-Wall lattice BW2m is in P2m). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f ∈ Pn (n ≥ 2) with coefficients restricted to the set {1, 2,..., n}. (1) Supported by the AT&T Labs Fellowship Program.