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On the integrality of nth roots of generating functions

Heninger, Nadia and Rains, E. M. and Sloane, N. J. A. (2006) On the integrality of nth roots of generating functions. Journal of Combinatorial Theory. Series A, 113 (8). pp. 1732-1745. ISSN 0097-3165. doi:10.1016/j.jcta.2006.03.018.

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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}.

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Additional Information:© 2006 Elsevier Inc. Received 13 September 2005, Available online 12 June 2006. Supported by the AT&T Labs Fellowship Program.
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Subject Keywords:Formal power series; Square roots of series; Fractional powers; Integer sequences; Theta series; Barnes–Wall lattices; E8 lattice; Leech lattice; Weight enumerators; BCH codes; Kerdock codes; Preparata codes; Reed–Muller codes
Issue or Number:8
Record Number:CaltechAUTHORS:20171027-091207506
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Official Citation:Nadia Heninger, E.M. Rains, N.J.A. Sloane, On the integrality of nth roots of generating functions, In Journal of Combinatorial Theory, Series A, Volume 113, Issue 8, 2006, Pages 1732-1745, ISSN 0097-3165, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:82726
Deposited By: Tony Diaz
Deposited On:27 Oct 2017 16:21
Last Modified:15 Nov 2021 19:52

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