On the integrality of nth roots of generating functions

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:113

Article

Heninger, Nadia ; Rains, E. M. ; Sloane, N. J. A.

关键词: formal power series;    square roots of series;    fractional powers;    integer sequences;    theta series;    Barnes-Wall lattices;    E-g lattice;    Leech lattice;    weight enumerators;    BCH codes;    Kerdock codes;    Preparata codes;    Reed-Muller codes;   

DOI  :  10.1016/j.jcta.2006.03.018

来源: Elsevier

【 摘 要 】

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 is an element of R (where R = 1 + xZ[x]) can be written as f = g(n) for g is an element of R, n >= 2. Let P-n := {g(n) vertical bar g is an element of R} and let mu(n) := n Pi(p vertical bar n) p. We show among other things that (i) for f is an element of R, f is an element of P-n double left right arrow f (mod mu(n)) is an element of P-n, and (ii) if f is an element of P-n, there is a unique g is an element of P-n with coefficients mod mu(n)/n such that f equivalent to g(n) (mod mu(n)). In particular, if f equivalent to 1 (mod mu(n)) then f is an element of 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(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 2(m) is in P-2r (and similarly that the theta series of the Barnes-Wall lattice BW2m 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 is an element of P-n (n >= 2) with coefficients restricted to the set {1, 2, ..., n}. (c) 2006 Elsevier Inc. All rights reserved.

【 授权许可】

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