Boolean constant degree functions on the slice are juntas

We show that a Boolean degree d function on the slice ([n] k ) = {(x1, . . . , xn) ∈ {0, 1} : ∑n i=1 xi = 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.