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On the size of homogeneous and of depth-four formulas with low individual degree

Kayal, N and Saha, C and Tavenas, S (2018) On the size of homogeneous and of depth-four formulas with low individual degree. In: Theory of Computing, 14 .

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Official URL: https://doi.org/10.4086/toc.2018.v014a016

Abstract

Let r � 1 be an integer. Let us call a polynomial f (x 1 , x 2 , �, x N ) � Fx a multi-r-ic polynomial if the degree of f with respect to any variable is at most r. (This generalizes the notion of multilinear polynomials.) We investigate the arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. We prove lower bounds for several subclasses of such circuits, including the following. 1. An N Ω(logN) lower bound against homogeneous multi-r-ic formulas (for an explicit multi-r-ic polynomial ( � ) on N variables). 2. An (n/r 1.1 ) Ωd/r lower bound against depth-four multi-r-ic circuits computing the polynomial IMM n,d corresponding to the product of d matrices of size n � n each. © 2018 Neerak Kayal, Chandan Saha, and Sébastien Tavenas.

Item Type: Journal Article
Additional Information: Copyright for this article belongs to
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Depositing User: Id for Latest eprints
Date Deposited: 20 Mar 2019 05:54
Last Modified: 20 Mar 2019 05:54
URI: http://eprints.iisc.ac.in/id/eprint/62047

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