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On defining integers and proving arithmetic circuit lower bounds
Citation key B-On-Defining-Integers-And-Proving-Arithmetic-Circuit-Lower-Bounds
Author Peter Bürgisser
Pages 81-103
Year 2009
Journal Computational Complexity
Volume 18
Note Preliminary version in Proc. of STACS 2007, 133-144, February 2007, Aachen
Abstract Let $\tau(n)$ 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 $n× n$ matrices having size polynomial in $n$, then $\tau(n!)$ is polynomially bounded in $łog n$. Under the same assumption on the permanent, we conclude that the Pochhammer-Wilkinson polynomials $\prod_k=1^n (X-k)$ and the Taylor approximations $\sum_k=0^n \frac1k! X^k$ and $\sum_k=1^n \frac1k X^k$ of $\exp$ and $łog$, respectively, can be computed by arithmetic circuits of size polynomial in $łog n$ (allowing divisions). This connects several so far unrelated conjectures in algebraic complexity.
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