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On the Size of Homogeneous and of Depth-Four Formulas with Low Individual Degree

Kayal, Neeraj and Saha, Chandan and Tavenas, Sebastien (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)) is an element of 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-Omega((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))(Omega)((root d/r)) lower bound against depth-four multi-r-ic circuits computing the polynomial IMMn,d corresponding to the product of d matrices of size n x n each. 3. A 2(Omega)((root N)) lower bound against depth-four multi-r-ic circuits computing an explicit multi-r-ic polynomial on N variables.

Item Type: Journal Article
Additional Information: Copyright of this article belongs to UNIV CHICAGO, DEPT COMPUTER SCIENCE
Keywords: complexity theory; lower bounds; algebraic complexity; arithmetic formulas; arithmetic circuits; partial derivatives
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Depositing User: Francis Jayakanth
Date Deposited: 10 Feb 2019 09:42
Last Modified: 10 Feb 2019 09:42
URI: http://eprints.iisc.ac.in/id/eprint/61361

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