A primer on the closure of algebraic complexity classes under factoring

TL;DR

This survey explores the closure properties of algebraic complexity classes under polynomial factorization, emphasizing techniques like Hensel lifting and Newton iteration and their implications for classes such as VP, VNP, and VBP.

Contribution

It organizes and analyzes known results on polynomial factorization within algebraic complexity classes, highlighting the roles of key techniques and discussing open problems.

Findings

Hensel lifting and Newton iteration are pivotal in polynomial factorization.

The paper clarifies the equivalence and application contexts of these techniques.

Several open problems in algebraic complexity remain unresolved.

Abstract

Polynomial factorization is a fundamental problem in computational algebra. Over the past half century, a variety of algorithmic techniques have been developed to tackle different variants of this problem. In parallel, algebraic complexity theory classifies polynomials into complexity classes based on their perceived `hardness'. This raises a natural question: Do these classes afford efficient factorization? In this survey, we revisit two pivotal techniques in polynomial factorization: Hensel lifting and Newton iteration. Though they are variants of the same theme, their distinct applications across the literature warrant separate treatment. These techniques have played an important role in resolving key factoring questions in algebraic complexity theory. We examine and organise the known results through the lens of these techniques to highlight their impact. We also discuss their equivalence while reflecting on how their use varies with the context of the problem. We focus on four prominent complexity classes: circuits of polynomial size ($\text{VP}_{\text{nb}}$), circuits with both polynomial size and degree (VP and its border $\overline{\text{VP}}$), verifier circuits of polynomial size and degree (VNP), and polynomial-size algebraic branching programs (VBP). We also examine more restricted models, such as formulas and bounded-depth circuits. Along the way, we list several open problems that remain unresolved.