A Note on Polynomial Identity Testing for Depth-3 Circuits

TL;DR

This paper presents a deterministic polynomial-time algorithm for identity testing of depth-3 arithmetic circuits with bounded fan-in, improving the efficiency of verifying polynomial identities in algebraic complexity.

Contribution

It introduces a deterministic algorithm for polynomial identity testing of depth-3 circuits with bounded fan-in, advancing the understanding of circuit complexity.

Findings

Deterministic polynomial-time identity testing for depth-3 circuits with bounded fan-in.

Algorithm runs in time $2^d ext{ poly}(n,s)$, where $d$ is the fan-in bound.

Applicable over fields $Q$ and $C$.

Abstract

Let $C$ be a depth-3 arithmetic circuit of size at most $s$, computing a polynomial $ f \in \mathbb{F}[x_1,\ldots, x_n] $ (where $\mathbb{F}$ = $\mathbb{Q}$ or $\mathbb{C}$) and the fan-in of the product gates of $C$ is bounded by $d$. We give a deterministic polynomial identity testing algorithm to check whether $f\equiv 0$ or not in time $ 2^d \text{ poly}(n,s) $.