# Lower bounds for depth-three circuits with equals and mod-gates

## Abstract

We say an integer polynomial *p*, on Boolean inputs, weakly *m*-represents a Boolean function *f* if *p* is non-constant and is zero (mod *m*) whenever *f* is zero. In this paper we prove that if a polynomial weakly *m*-represents the Mod_{q} function on *n* inputs, where *q* and *m* are relatively prime and *m* is otherwise arbitrary, then the degree of the polynomial is *Ω(n)*. This generalizes previous results of Barrington, Beigel and Rudich [BBR] and Tsai [Tsai], which held only for constant (or slowly growing) *m*. The proof technique given here is quite different and somewhat simpler. We use a method in which the inputs are represented as complex *q*^{th} roots of unity (following Barrington and Straubing [BS]). The representation is used to take advantage of a variant of the inverse Fourier transform and elementary properties of the algebraic integers. As a corollary of the main theorem and the proof of Toda's theorem, if *q, p* are distinct primes, any depth-three circuit which computes the Mod_{q} function, and consists of an equals gate at the output, Mod_{p}-gates at the next level, and AND-gates of small fan-in at the inputs, must be of exponential size. In terms of Turing machine complexity classes, there is an oracle *A* such that \(Mod_q P^A \nsubseteq C_ = P^{Mod_p P^A }\).

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