On the number of relevant variables for discrete functions
V. N. Potapov
TL;DR
The work investigates how the number of relevant variables in discrete $q$-ary functions is controlled by generalized degrees deg_0, deg_1, and deg_2. It develops a Fourier–Hadamard spectral framework on $\mathbb{Z}_q^n$ and leverages average sensitivity on Cartesian products of cycles to establish new upper bounds for two-valued and three-valued functions that depend on deg_1 and deg_2 in addition to deg_0. The results extend and refine classical bounds for Boolean functions (e.g., Nisan–Szegedy) and provide concrete bounds for classes of functions, including constructions showing where the new bounds improve existing ones. The methods tie together degree notions, spectral graph theory, and colorings of Lee-type graphs, offering a unifying approach to relate function degree and the count of essential variables.
Abstract
We consider various definitions of degrees of discrete functions and establish relations between the number of relevant (essential) variables and degrees of two- and three-valued functions. Keywords: relevant variable, sensitivity, degree of Boolean function.