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Generalized polynomials and hyperplane functions in (Z/pkZ)n

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https://doi.org/10.5070/C65165028Creative Commons 'BY' version 4.0 license
Abstract

For p prime, let Hn be the linear span of indicator functions of hyperplanes in (Z/pkZ)n. We establish new upper bounds on the dimension of Hn over Z/pZ, or equivalently, on the rank of point-hyperplane incidence matrices in (Z/pkZ)n over Z/pZ. Our proof is based on a variant of the polynomial method using binomial coefficients in Z/pkZ as generalized polynomials. We also establish additional necessary conditions for a function on (Z/pkZ)n to be an element of Hn.

Mathematics Subject Classifications: 05B20, 05B25, 05A10

Keywords: Hyperplanes, generalized polynomials, binomial coefficients

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