Deterministic Algorithms for Low Degree Factors of Constant Depth Circuits

TL;DR

This paper presents a deterministic, subexponential time algorithm for finding low-degree factors of multivariate polynomials represented by constant depth circuits, with improved performance on sparse polynomials, advancing algebraic circuit factorization techniques.

Contribution

It introduces a novel deterministic algorithm for low-degree polynomial factorization over constant depth circuits, connecting polynomial identity testing with factorization in a new way.

Findings

Deterministic subexponential algorithm for low-degree factors

Quasipolynomial time for sparse polynomials

Enhanced connection between PIT and polynomial factorization

Abstract

For every constant $d$, we design a subexponential time deterministic algorithm that takes as input a multivariate polynomial $f$ given as a constant depth algebraic circuit over the field of rational numbers, and outputs all irreducible factors of $f$ of degree at most $d$ together with their respective multiplicities. Moreover, if $f$ is a sparse polynomial, then the algorithm runs in quasipolynomial time. Our results are based on a more fine-grained connection between polynomial identity testing (PIT) and polynomial factorization in the context of constant degree factors and rely on a clean connection between divisibility testing of polynomials and PIT due to Forbes and on subexponential time deterministic PIT algorithms for constant depth algebraic circuits from the recent work of Limaye, Srinivasan and Tavenas.