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On Defining Integers in the Counting Hierarchy and Proving Arithmetic Circuit Lower Bounds

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4393))

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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 for computing the permanent of n by n matrices having size polynomial in n, then τ(n!) is polynomially bounded in logn. Under the same assumption on the permanent, we conclude that the Pochhammer-Wilkinson polynomials \(\prod_{k=1}^n (X-k)\) and the Taylor approximations \(\sum_{k=0}^n \frac1{k!} X^k\) and \(\sum_{k=1}^n \frac1{k} X^k\) of exp and log, respectively, can be computed by arithmetic circuits of size polynomial in logn (allowing divisions). This connects several so far unrelated conjectures in algebraic complexity.

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  1. Dept. of Mathematics, University of Paderborn, D-33095 Paderborn, Germany

    Peter Bürgisser

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  1. Peter Bürgisser
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Wolfgang Thomas Pascal Weil

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Bürgisser, P. (2007). On Defining Integers in the Counting Hierarchy and Proving Arithmetic Circuit Lower Bounds. In: Thomas, W., Weil, P. (eds) STACS 2007. STACS 2007. Lecture Notes in Computer Science, vol 4393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70918-3_12

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  • DOI: https://doi.org/10.1007/978-3-540-70918-3_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-70917-6

  • Online ISBN: 978-3-540-70918-3

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