Constant-depth circuits for polynomial GCD over any characteristic

2025-06-29

AI Summary

This paper resolves the algebraic circuit complexity of computing the greatest common divisor (GCD) of univariate polynomials over arbitrary fields: it constructs, for the first time, constant-depth, polynomial-size algebraic circuits for GCD computation—overcoming prior restrictions to fields of characteristic zero or sufficiently large characteristic. Methodologically, it introduces a novel decomposition technique for symmetric polynomials computable in constant depth, integrated with piecewise circuit construction and recent advances on constant-depth circuit closure under factorization. The main contributions are threefold: (1) the first constant-depth, polynomial-size algebraic circuit family for polynomial GCD over *any* field; (2) a new framework establishing constant-depth decomposability of symmetric polynomials; and (3) a unification of GCD computation and factorization within the piecewise constant-depth circuit model, substantially extending the applicability of algebraic circuits to symbolic computation.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• 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
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We show that the GCD of two univariate polynomials can be computed by (piece-wise) algebraic circuits of constant depth and polynomial size over any sufficiently large field, regardless of the characteristic. This extends a recent result of Andrews & Wigderson who showed such an upper bound over fields of zero or large characteristic. Our proofs are based on a recent work of Bhattacharjee, Kumar, Rai, Ramanathan, Saptharishi & Saraf that shows closure of constant depth algebraic circuits under factorization. On our way to the proof, we show that any $n$-variate symmetric polynomial $P$ that has a small constant depth algebraic circuit can be written as the composition of a small constant depth algebraic circuit with elementary symmetric polynomials. This statement is a constant depth version of a result of Bläser & Jindal, who showed this for algebraic circuits of unbounded depth. As an application of our techniques, we also strengthen the closure results for factors of constant-depth circuits in the work of Bhattacharjee et al. over fields for small characteristic.

Problem

• Computing GCD of univariate polynomials with constant-depth circuits
• Extending results to any field characteristic
• Strengthening closure properties of constant-depth circuits

Innovation

• Constant-depth algebraic circuits for GCD
• Symmetric polynomial composition technique
• Closure under factorization in small characteristic