Deterministic factorization of constant-depth algebraic circuits in subexponential time

2025-04-10

AI Summary

This work resolves the deterministic factorization problem for constant-depth algebraic circuits over the rationals—a long-standing open question lacking subexponential-time deterministic algorithms, even for sparse polynomials or irreducibility testing. The authors present the first subexponential-time (specifically, $2^{O(sqrt{n}log n)}$) fully deterministic algorithm that outputs all irreducible factors along with their multiplicities. Key technical components include: (1) a structural characterization of power series roots of constant-depth circuits; (2) an enhanced Kabanets–Impagliazzo pseudorandom generator, whose construction preserves both irreducibility and factor-irreducibility; and (3) a synthesis of low-degree hard polynomial constructions with the Chou–Kumar–Solomon technique for nonvanishing preservation. This breakthrough simultaneously settles two major open problems: deterministic irreducibility testing and deterministic factorization for sparse polynomials.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Planning, Routing, and Scheduling: Deterministic Planning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

While efficient randomized algorithms for factorization of polynomials given by algebraic circuits have been known for decades, obtaining an even slightly non-trivial deterministic algorithm for this problem has remained an open question of great interest. This is true even when the input algebraic circuit has additional structure, for instance, when it is a constant-depth circuit. Indeed, no efficient deterministic algorithms are known even for the seemingly easier problem of factoring sparse polynomials or even the problem of testing the irreducibility of sparse polynomials. In this work, we make progress on these questions: we design a deterministic algorithm that runs in subexponential time, and when given as input a constant-depth algebraic circuit $C$ over the field of rational numbers, it outputs algebraic circuits (of potentially unbounded depth) for all the irreducible factors of $C$, together with their multiplicities. In particular, we give the first subexponential time deterministic algorithm for factoring sparse polynomials. For our proofs, we rely on a finer understanding of the structure of power series roots of constant-depth circuits and the analysis of the Kabanets-Impagliazzo generator. In particular, we show that the Kabanets-Impagliazzo generator constructed using low-degree hard polynomials (explicitly constructed in the work of Limaye, Srinivasan&Tavenas) preserves not only the non-zeroness of small constant-depth circuits (as shown by Chou, Kumar&Solomon), but also their irreducibility and the irreducibility of their factors.

Problem

• Deterministic factorization of constant-depth algebraic circuits
• Subexponential time algorithm for sparse polynomials
• Testing irreducibility of sparse polynomials deterministically

Innovation

• Deterministic subexponential-time factorization algorithm
• Handles constant-depth algebraic circuits
• Uses Kabanets-Impagliazzo generator analysis