Constant-depth circuits for polynomial GCD over any characteristic

TL;DR

This paper demonstrates that the GCD of two univariate polynomials can be computed using constant-depth algebraic circuits over any large field, extending previous results to all characteristics and providing new insights into polynomial factorization.

Contribution

It introduces a method to compute polynomial GCDs with constant-depth circuits over any characteristic, generalizing prior zero or large characteristic results and strengthening closure properties for factors.

Findings

GCD computation can be achieved with constant-depth circuits over any large field.

Symmetric polynomials with small circuits can be expressed via elementary symmetric polynomials.

Strengthens closure results for factors of constant-depth circuits in small characteristic fields.

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 Bl\"{a}ser & 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.