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Polynomial enumeration of multidimensional lattices

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Abstract

Denoting the nonnegative (resp. signed) integers byN (resp.Z) and the real numbers byR, letK ⊂ R m andf: R m → R. Thenf is astoring function (resp.packing function) onK wheneverf|(Z m ⊒ K) is an injection into (resp. bijection onto)N. Unit translations gm of some P. Chowla [1961] polynomials are packing functions on the correspondingN m, and all compositions of these polynomials yield further packing functions on variousN r. We study this accessible family of packing functions, using standard properties ofordered trees to classify all those compositions, up to a simple equivalence, which define polynomial packing functions on eachN m. The numberc(m) of equivalence classes is an exponentially growing function for largem, whence the uniqueness conjecture of our prior two-dimensional work has no counterpart for largerm. We obtain the admissible degrees for composition polynomials inm variables; we describe the tre structures for all such polynomials with extremal degrees. Them-variable polynomials of least degree form a rather irregular numbera(m) of equivalence classes. Density considerations give some degree constraints on ageneral polynomial packing function whose domainK is the topological closure of a nonvoid open cone.

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  1. Mathematical Sciences Department, IBM T. J. Watson Research Center, P. O. Box 218, 10598, Yorktown Heights, N.Y., USA

    John S. Lew

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  1. John S. Lew
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Lew, J.S. Polynomial enumeration of multidimensional lattices. Math. Systems Theory 12, 253–270 (1978). https://doi.org/10.1007/BF01776577

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  • DOI: https://doi.org/10.1007/BF01776577

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