Algorithmica

, Volume 76, Issue 4, pp 890–909 | Cite as

Building Above Read-Once Polynomials: Identity Testing and Hardness of Representation

  • Meena Mahajan
  • B. V. Raghavendra Rao
  • Karteek Sreenivasaiah
Article

Abstract

Polynomial Identity Testing (PIT) algorithms have focussed on polynomials computed either by small alternation-depth arithmetic circuits, or by read-restricted formulas. Read-once polynomials (ROPs) are computed by read-once formulas (ROFs) and are the simplest of read-restricted polynomials. Building structures above these, we show the following: (1) a deterministic polynomial-time non-black-box PIT algorithm for \(\sum ^{(2)}\times \prod \times \mathsf{ROF}\). (2) Weak hardness of representation theorems for sums of powers of constant-free ROPs and for \(\mathsf{ROF}\)s of the form \(\sum \times \prod \times \sum \). (3) A partial characterization of multilinear monotone constant-free ROPs.

Keywords

Polynomial Identity Testing Algebraic algorithms Arithmetic circuits 

Notes

Acknowledgments

The authors gratefully acknowledge Amir Shpilka’s pointer regarding Theorem 2, when he and the first author were at the Dagstuhl Seminar 14121 on Computational Complexity of Discrete Problems. The authors are grateful to anonymous reviewers for their careful reading of the manuscript, several comments to improve readability, and for pointing out why the converse of Lemma 9 fails.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Meena Mahajan
    • 1
  • B. V. Raghavendra Rao
    • 2
  • Karteek Sreenivasaiah
    • 3
  1. 1.The Institute of Mathematical SciencesChennaiIndia
  2. 2.Indian Institute of Technology MadrasChennaiIndia
  3. 3.Max Planck Institute for InformaticsSaarbrückenGermany

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