On perfect hashing of numbers with sparse digit representation via multiplication by a constant
Consider the set of vectors over a field having non-zero coefficients only in a fixed sparse set and multiplication defined by convolution, or the set of integers having non-zero digits (in some base b b ) in a fixed sparse set. We show the existence of an optimal (or almost-optimal, in the latter case) magic multiplier constant that provides a perfect hash function which transfers the information from the given sparse coefficients into consecutive digits. Studying the convolution case we also obtain a result of non-degeneracy for Schur functions as polynomials in the elementary symmetric functions in positive characteristic.
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