Factors of low individual degree polynomials

Abstract

Kaltofen (Randomness in computation, vol 5, pp 375---412, 1989) proved the remarkable fact that multivariate polynomial factorization can be done efficiently, in randomized polynomial time. Still, more than twenty years after Kaltofen's work, many questions remain unanswered regarding the complexity aspects of polynomial factorization, such as the question of whether factors of polynomials efficiently computed by arithmetic formulas also have small arithmetic formulas, asked in Kopparty et al. (2014), and the question of bounding the depth of the circuits computing the factors of a polynomial. We are able to answer these questions in the affirmative for the interesting class of polynomials of bounded individual degrees, which contains polynomials such as the determinant and the permanent. We show that if $${P(x_{1},\ldots,x_{n})}$$P(x1,?,xn) is a polynomial with individual degrees bounded by r that can be computed by a formula of size s and depth d, then any factor $${f(x_{1},\ldots, x_{n})}$$f(x1,?,xn) of $${P(x_{1},\ldots,x_{n})}$$P(x1,?,xn) can be computed by a formula of size $${\textsf{poly}((rn)^{r},s)}$$poly((rn)r,s) and depth d + 5. This partially answers the question above posed in Kopparty et al. (2014), who asked if this result holds without the dependence on r. Our work generalizes the main factorization theorem from Dvir et al. (SIAM J Comput 39(4):1279---1293, 2009), who proved it for the special case when the factors are of the form $${f(x_{1}, \ldots, x_{n}) \equiv x_{n} - g(x_{1}, \ldots, x_{n-1})}$$f(x1,?,xn)?xn-g(x1,?,xn-1). Along the way, we introduce several new technical ideas that could be of independent interest when studying arithmetic circuits (or formulas).