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On Defining Integers And Proving Arithmetic Circuit Lower Bounds

Abstract.

Let τ(n) denote the minimum number of arithmetic operations sufficient to build the integer n from the constant 1. We prove that if there are arithmetic circuits of size polynomial in n for computing the permanent of n by n matrices, then τ(n!) is polynomially bounded in log n. Under the same assumption on the permanent, we conclude that the Pochhammer–Wilkinson polynomials \(\Pi^{n}_{k=1}(X - k)\) and the Taylor approximations \(\Sigma^{n}_{k=0}\frac{1}{k!}X^{k}\) and \(\Sigma^{n}_{k=1}\frac{1}{k}X^{k}\) of exp and log, respectively, can be computed by arithmetic circuits of size polynomial in log n (allowing divisions). This connects several so far unrelated conjectures in algebraic complexity.

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Correspondence to Peter Bürgisser.

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Manuscript received 25 August 2006

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Bürgisser, P. On Defining Integers And Proving Arithmetic Circuit Lower Bounds. comput. complex. 18, 81–103 (2009). https://doi.org/10.1007/s00037-009-0260-x

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Keywords.

  • Algebraic complexity
  • permanent
  • factorials
  • integer roots of univariate polynomials

Subject classification.

  • Primary 68Q17
  • Secondary 11D45