Boolean constant degree functions on the slice are juntas

TL;DR

This paper proves that Boolean degree d functions on large slices are juntas, extending classical hypercube results, and shows the maximum number of influential coordinates is the same on slices and hypercubes.

Contribution

It generalizes Nisan and Szegedy's hypercube junta result to slices and establishes the maximum influence coordinate count equivalence.

Findings

Boolean degree d functions on large slices are juntas

Maximum influential coordinates are the same on slices and hypercubes

Extension of classical hypercube results to combinatorial slices

Abstract

We show that a Boolean degree $d$ function on the slice $\binom{[n]}{k} = \{ (x_1,\ldots,x_n) \in \{0,1\} : \sum_{i=1}^n x_i = k \}$ is a junta, assuming that $k,n-k$ are large enough. This generalizes a classical result of Nisan and Szegedy on the hypercube. Moreover, we show that the maximum number of coordinates that a Boolean degree $d$ function can depend on is the same on the slice and the hypercube.