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computational complexity

, Volume 10, Issue 1, pp 1–27 | Cite as

Depth-3 arithmetic circuits over fields of characteristic zero

  • A. Shpilka
  • A. Wigderson

Abstract.

In this paper we prove quadratic lower bounds for depth-3 arithmetic circuits over fields of characteristic zero. Such bounds are obtained for the elementary symmetric functions, the (trace of) iterated matrix multiplication, and the determinant. As corollaries we get the first nontrivial lower bounds for computing polynomials of constant degree, and a gap between the power of depth-3 arithmetic circuits and depth-4 arithmetic circuits. We also give new shorter formulae of constant depth for the elementary symmetric functions.¶The main technical contribution relates the complexity of computing a polynomial in this model to the wealth of partial derivatives it has on every affine subspace of small co-dimension. Lower bounds for related models utilize an algebraic analog of the Neciporuk lower bound on Boolean formulae.

Keywords. Depth-3 circuits, symmetric functions, lower bounds, partial derivatives. 

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Copyright information

© Birkhäuser Verlag, Basel 2001

Authors and Affiliations

  • A. Shpilka
    • 1
  • A. Wigderson
    • 2
  1. 1.Department of Computer Science, Hebrew University, Jerusalem, Israel 91904, e-mail: amirs@cs.huji.ac.il, http://www.cs.huji.ac.il/~amirs/IL
  2. 2.Department of Computer Science, Hebrew University, Jerusalem, Israel 91904, e-mail: avi@cs.huji.ac.il, http://www.cs.huji.ac.il/~avi/IL

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