Complexity Theory Column 88

Abstract

Algebraic complexity theory studies the complexity of computing (multivariate) polynomials efficiently using algebraic circuits. This succinct representation leads to fundamental algorithmic challenges such as the polynomial identity testing (PIT) problem (decide nonzeroness of the computed polynomial) and the polynomial factorization problem (compute succinct representations of the factors of the circuit). While the Schwartz-Zippel-DeMillo-Lipton Lemma [Sch80,Zip79,DL78] gives an easy randomized algorithm for PIT, randomized algorithms for factorization require more ideas as given by Kaltofen [Kal89]. However, even derandomizing PIT remains a fundamental problem in understanding the power of randomness. In this column, we survey the factorization problem, discussing the algorithmic ideas as well as the applications to other problems. We then discuss the challenges ahead, in particular focusing on the goal of obtaining deterministic factoring algorithms. While deterministic PIT algorithms have been developed for various restricted circuit classes, there are very few corresponding factoring algorithms. We discuss some recent progress on the divisibility testing problem (test if a given polynomial divides another given polynomial) which captures some of the difficulty of factoring. Along the way we attempt to highlight key challenges whose solutions we hope will drive progress in the area.

What Can (and Can't) we Do with Sparse Polynomials?

“…This question is already addressed in some other surveys such as [22,26,54]. Because of this, and since this is the area in which the speaker has the least expertise, we give only a very cursory overview of the accomplishments and challenges here.…”

Untitled

“…Specifically, von zur Gathen and Kaltofen [30] gave an explicit s-sparse polynomial (over any field) which has a factor with s Ω(log s) monomials, and Volkovich [85] gave, for a prime p, an explicit n-variate n-sparse polynomial of degree-p which in characteristic p has a factor with n+p−2 n−1 monomials (an exponential separation for p ≥ poly(n)). We refer the reader to the survey of Forbes and Shpilka [28] for more on the challenges in factoring small algebraic circuits.…”