m-ary Sensitivity Conjecture

Establish a polynomial upper bound on the polynomial degree of an m-ary function f: T^n → T (with |T| = m) in terms of its sensitivity s(f); specifically, determine whether for every integer m there exists a constant c such that deg(f) ∈ O(s(f)^c) for every m-ary function f.

Background

The classical sensitivity conjecture for Boolean functions was resolved by Huang via a graph-theoretic reformulation. This paper extends sensitivity, block sensitivity, and degree to m-ary functions f: T^n → T and proves the upper bound s(f) = O((deg f)^2). The challenging direction is to bound deg(f) polynomially by s(f), prompting the generalization stated here.

The authors also develop an equivalence theorem connecting this conjecture to low-degree partitions of Hamming graphs, and they show a quadratic separation (deg(f) ∈ Ω(s(f)^2)) for p-ary functions with p prime, implying that any valid exponent c must be at least 2.