@inproceedings{246c57a03fbb4080a58849ffb3c19871,

title = "On the complexity of Boolean functions in different characteristics",

abstract = "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(n1-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.",

keywords = "Boolean functions, Polynomials",

author = "Parikshit Gopalan and Shachar Lovett and Amir Shpilka",

year = "2009",

doi = "10.1109/CCC.2009.14",

language = "אנגלית",

isbn = "9780769537177",

series = "Proceedings of the Annual IEEE Conference on Computational Complexity",

pages = "173--183",

booktitle = "Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009",

note = "null ; Conference date: 15-07-2009 Through 18-07-2009",

}