Low-degree functions without non-essential arguments

TL;DR

This paper constructs special low-degree functions on Hamming graphs with a focus on the exponential relationship between the number of essential arguments and the function's degree, revealing new structural properties.

Contribution

It introduces a method to construct perfect colorings without non-essential arguments with exponential dependence on the quotient matrix's off-diagonal part.

Findings

Constructed perfect colorings with exponential dependence on parameters

Demonstrated unbalanced Boolean functions with exponentially many essential arguments

Revealed structural properties of low-degree functions on Hamming graphs

Abstract

For the Hamming graph $H(n,q)$, where a $q$ is a constant prime power and $n$ grows, we construct perfect colorings without non-essential arguments such that $n$ depends exponentially on the off-diagonal part of the quotient matrix. In particular, we construct unbalanced Boolean ($q=2$) functions such that the number of essential arguments depends exponentially on the degree of the function.