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.
https://resolver.caltech.edu/CaltechAUTHORS:20171027-091207506

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## 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}.

Item Type: | Article | ||||||||||||
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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. | ||||||||||||

Funders: |
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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 | ||||||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20171027-091207506 | ||||||||||||

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, https://doi.org/10.1016/j.jcta.2006.03.018. (http://www.sciencedirect.com/science/article/pii/S0097316506000677) | ||||||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||

ID Code: | 82726 | ||||||||||||

Collection: | CaltechAUTHORS | ||||||||||||

Deposited By: | Tony Diaz | ||||||||||||

Deposited On: | 27 Oct 2017 16:21 | ||||||||||||

Last Modified: | 03 Oct 2019 18:57 |

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