## 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: ^{o} 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 at most one characteristic. This result is essentially tight as there exist functions that have degree log n in every characteristic. ^{o} If f has degree d = o(log n) modulo p, then it cannot be computed correctly on more than 1 - p^{-O(d)} fraction of the hypercube by polynomials of degree, 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 AC_{0}[q] circuits. This gives a lower bound for a broad and natural class of functions.

Original language | English |
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Pages (from-to) | 235-263 |

Number of pages | 29 |

Journal | Computational Complexity |

Volume | 19 |

Issue number | 2 |

DOIs | |

State | Published - May 2010 |

Externally published | Yes |

## Keywords

- Bounded depth circuits
- Low degree polynomials
- Lower bounds