Arithmetic circuit complexity of division and truncation

Abstract

Given polynomials f, g, h ∈ F[x1, ..., xn] such that f = g/h, where both g and h are computable by arithmetic circuits of size s, we show that f can be computed by a circuit of size poly(s, deg(h)). This solves a special case of division elimination for high-degree circuits (Kaltofen'87 & WACT'16). The result is an exponential improvement over Strassen's classic result (Strassen'73) when deg(h) is poly(s) and deg(f) is exp(s), since the latter gives an upper bound of poly(s, deg(f)). Further, we show that any univariate polynomial family (fd)d, defined by the initial segment of the power series expansion of rational function gd(x)/hd(x) up to degree d (i.e. fd = gd/hd mod xd+1), where circuit size of g is sd and degree of gd is at most d, can be computed by a circuit of size poly(sd, deg(hd), log d). We also show a hardness result when the degrees of the rational functions are high (i.e. Ω(d)), assuming hardness of the integer factorization problem. Finally, we extend this conditional hardness to simple algebraic functions as well, and show that for every prime p, there is an integral algebraic power series with its minimal polynomial satisfying a degree p polynomial equation, such that its initial segment is hard to compute unless integer factoring is easy, or a multiple of n! is easy to compute. Both, integer factoring and computation of multiple of n!, are believed to be notoriously hard. In contrast, we show examples of transcendental power series whose initial segments are easy to compute.