On the degree of symmetric functions on the Boolean cube

In this paper we study the degree of non-constant symmetric functions f : {0, 1}n → {0, 1, . . . , c}, where c ∈ N, when represented as polynomials over the real numbers. We show that as long as c < n it holds that deg(f) = Ω(n). As we can have deg(f) = 1 when c = n, our result shows a surprising threshold phenomenon. The question of lower bounding the degree of symmetric functions on the Boolean cube was previously studied by von zur Gathen and Roche [GR97] who showed the lower bound deg(f) ≥ n+1 c+1 and so our result greatly improves this bound. When c = 1, namely the function maps the Boolean cube to {0, 1}, we show that if n = p2, when p is a prime, then deg(f) ≥ n − √ n. This slightly improves the previous bound of [GR97] for this case.

• Type: Journal
• Year: 2010
• Where: ECCC
• Authors: Gil Cohen, Amir Shpilka
• Keywords: ECCC 2010, Non-constant Symmetric Functions, Surprising Threshold Phenomenon, Symmetric Functions