On the complexity of Boolean functions in different characteristics

Every Boolean function on n variables can be expressed as a unique multivariate polynomial modulo p for every prime p. in this work, we study how the degree of a function in one characteristic affects its complexity in other characteristics. We establish the following general principle: functions with low degree modulo p must have high complexity in every other characteristic q. More precisely, we show the following results about Boolean functions f : {0, 1}ⁿ → {0, 1} which depend on all n variables, and distinct primes p, q: • If f has degree o(log n) modulo p, then it must have degree Ω(n^1-o(1)) modulo q. Thus a Boolean function has degree o(log n) in only one characteristic. This result is essentially tight as there exist functions that have degree log n in every characteristic. • If f has degree d = o(log n) modulo p, it cannot be computed correctly on more than 1 - p ^-O(d)fraction of the hypercube by polynomials of degree n ^{1/2 - ε} modulo q. As a corollary of the above results it follows that if f has degree o(log n) modulo p, then it requires super-polynomial size AC₀[q] circuits. This gives a lower bound for a broad and natural class of functions.