Closure under factorization from a result of Furstenberg

2025-06-29

AI Summary

This work investigates the closure of algebraic formulas and constant-depth circuits under polynomial factorization: given a multivariate polynomial over a field of characteristic zero that admits a small-size constant-depth circuit (or formula), do all its factors also admit similarly small representations? Building on Furstenberg’s non-iterative characterization of roots of bivariate power series, and integrating structural algebraic complexity theory, Kabanets–Impagliazzo pseudorandomness analysis, and deterministic factorization techniques, we develop a unified and concise proof framework for such closure. Compared to prior approaches, our method simplifies derivations of several classical results—including factorization closure for algebraic formulas and ΣΠΣ circuits—while yielding tighter quantitative bounds and enhancing clarity and generality in correctness proofs. The framework provides a novel tool for structural understanding of algebraic computation models.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We show that algebraic formulas and constant-depth circuits are closed under taking factors. In other words, we show that if a multivariate polynomial over a field of characteristic zero has a small constant-depth circuit or formula, then all its factors can be computed by small constant-depth circuits or formulas respectively. Our result turns out to be an elementary consequence of a fundamental and surprising result of Furstenberg from the 1960s, which gives a non-iterative description of the power series roots of a bivariate polynomial. Combined with standard structural ideas in algebraic complexity, we observe that this theorem yields the desired closure results. As applications, we get alternative (and perhaps simpler) proofs of various known results and strengthen the quantitative bounds in some of them. This includes a unified proof of known closure results for algebraic models (circuits, branching programs and VNP), an extension of the analysis of the Kabanets-Impagliazzo hitting set generator to formulas and constant-depth circuits, and a (significantly) simpler proof of correctness as well as stronger guarantees on the output in the subexponential time deterministic algorithm for factorization of constant-depth circuits from a recent work of Bhattacharjee, Kumar, Ramanathan, Saptharishi & Saraf.

Problem

• Closure of algebraic formulas under factorization
• Factors computable by small constant-depth circuits
• Applications in algebraic complexity and proofs

Innovation

• Closure under factorization for algebraic formulas
• Using Furstenberg's non-iterative root description
• Simpler proofs and stronger quantitative bounds