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

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.

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Notes

  1. 1.

    A parse tree is a sub-formula of F (i) containing the output gate, (ii) including, for every included \(\vee \) gate, exactly one child, and (iii) including, for every included \(\wedge \) gate, both its children.

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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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Correspondence to B. V. Raghavendra Rao.

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Partially supported by the Indo-German Max Planck Center for Computer Science (IMPECS).

Karteek Sreenivasaiah: Much of this work was done while the author was working in The Institute of Mathematical Sciences, Chennai, India.

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Mahajan, M., Rao, B.V.R. & Sreenivasaiah, K. Building Above Read-Once Polynomials: Identity Testing and Hardness of Representation. Algorithmica 76, 890–909 (2016). https://doi.org/10.1007/s00453-015-0101-z

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Keywords

  • Polynomial Identity Testing
  • Algebraic algorithms
  • Arithmetic circuits