Abstract

In this paper we show that black-box polynomial identity testing for noncommutative polynomials f∈𝔽⟨z1,z2,…,zn⟩ of degree D and sparsity t, can be done in randomized (n,logt,logD) time. As a consequence, given a circuit C of size s computing a polynomial f∈𝔽⟨ z1,z2,…,zn⟩ with at most t non-zero monomials, then testing if f is identically zero can be done by a randomized algorithm with running time polynomial in s and n and logt. This makes significant progress on a question that has been open for over ten years. Our algorithm is based on automata-theoretic ideas that can efficiently isolate a monomial in the given polynomial. In particular, we carry out the monomial isolation using nondeterministic automata. In general, noncommutative circuits of size s can compute polynomials of degree exponential in s and number of monomials double-exponential in s. In this paper, we consider a natural class of homogeneous noncommutative circuits, that we call +-regular circuits, and give a white-box polynomial time deterministic polynomial identity test. These circuits can compute noncommutative polynomials with number of monomials double-exponential in the circuit size. Our algorithm combines some new structural results for +-regular circuits with known results for noncommutative ABP identity testing, rank bound of commutative depth three identities, and equivalence testing problem for words. Finally, we consider the black-box identity testing problem for depth three +-regular circuits and give a randomized polynomial time identity test. In particular, we show if f∈𝔽Z⟩ is a nonzero noncommutative polynomial computed by a depth three +-regular circuit of size s, then f cannot be a polynomial identity for the matrix algebra 𝕄s(𝔽) when 𝔽 is sufficiently large depending on the degree of f.

Highlights

  • Noncommutative computation, introduced in complexity theory by Hyafil [12] and Nisan [19], is a central field of algebraic complexity theory

  • We show if f is a nonzero noncommutative polynomial in n variables over the field F computed by a depth-3 +-regular circuit of size s, f cannot be a polynomial identity for the matrix algebra Ms(F) when F is sufficiently large depending on the degree of f

  • Bogdanov and Wee [8] have shown a randomized polynomial-time algorithm when the degree of the noncommutative circuit C is polynomially bounded in s and n

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Summary

Introduction

Noncommutative computation, introduced in complexity theory by Hyafil [12] and Nisan [19], is a central field of algebraic complexity theory. Bogdanov and Wee [8] have shown a randomized polynomial-time algorithm when the degree of the noncommutative circuit C is polynomially bounded in s and n. For any field F, a nonzero noncommutative polynomial P ∈ F Z of degree ≤ 2d − 1 is not a polynomial identity for the matrix algebra Md(F). If we do random substitutions from an extension field of F of size at least 4d, we get a randomized polynomial identity testing algorithm, with error probability at most 1/2, by the Polynomial Identity Lemma2 [20, 26, 18, 9, 27, 28] The problem with this approach for general noncommutative circuits (whose degree can be 2s) is that the dimension of the matrices grows linearly with the degree of the polynomial. Forbes-Shpilka [11] and Agrawal et al [1] have given a quasi-polynomial-time black-box algorithm for small degree noncommutative ABPs

Main results
Outline of the proofs
Preliminaries
The Polynomial Identity Lemma
Black-box PIT for polynomials of exponential degree and sparsity
A deterministic PIT for regular circuits
Projected polynomials
The black-box identity test
Findings
Conclusion
Full Text
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