New results on Noncommutative and Commutative Polynomial Identity Testing

TL;DR

This paper introduces a deterministic identity testing method for noncommutative polynomials using automata theory, and extends the approach to algebraic branching program reconstruction and commutative rings.

Contribution

It presents a novel automata-based deterministic identity test for noncommutative polynomials and applies it to reconstruct algebraic branching programs in polynomial time.

Findings

Efficient deterministic identity test for noncommutative polynomials

Polynomial-time reconstruction of algebraic branching programs

Extension of identity testing to polynomials over finite commutative rings

Abstract

Using ideas from automata theory we design a new efficient (deterministic) identity test for the \emph{noncommutative} polynomial identity testing problem (first introduced and studied in \cite{RS05,BW05}). We also apply this idea to the reconstruction of black-box noncommuting algebraic branching programs. Assuming the black-box model allows us to query the ABP for the output at any given gate, we can reconstruct an (equivalent) ABP in deterministic polynomial time. Finally, we explore commutative identity testing when the coefficients of the input polynomial come from an arbitrary finite commutative ring with unity.