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# An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

Kayal, Neeraj and Limaye, Nutan and Saha, Chandan and Srinivasan, Srikanth (2014) An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas. In: 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS), OCT 18-21, 2014, Microsoft Res New England, Philadelphia, PA, pp. 61-70. PDF 2014_IEEE_Ann_Sym_Fou_Com_Sci61_2014.pdf - Published Version Restricted to Registered users only Download (289kB) | Request a copy
Official URL: http://dx.doi.org/10.1109/FOCS.2014.15

## Abstract

We show here a 2(Omega(root d.log N)) size lower bound for homogeneous depth four arithmetic formulas. That is, we give an explicit family of polynomials of degree d on N variables (with N = d(3) in our case) with 0, 1-coefficients such that for any representation of a polynomial f in this family of the form f = Sigma(i) Pi(j) Q(ij), where the Q(ij)'s are homogeneous polynomials (recall that a polynomial is said to be homogeneous if all its monomials have the same degree), it must hold that Sigma(i,j) (Number of monomials of Q(ij)) >= 2(Omega(root d.log N)). The above mentioned family, which we refer to as the Nisan-Wigderson design-based family of polynomials, is in the complexity class VNP. Our work builds on the recent lower bound results 1], 2], 3], 4], 5] and yields an improved quantitative bound as compared to the quasi-polynomial lower bound of 6] and the N-Omega(log log (N)) lower bound in the independent work of 7].

Item Type: Conference Proceedings Copy right for this article belongs to the IEEE, 345 E 47TH ST, NEW YORK, NY 10017 USA Arithmetic circuits; shifted partial derivatives; lower bounds Division of Electrical Sciences > Computer Science & Automation Id for Latest eprints 06 Jan 2016 05:44 06 Jan 2016 05:44 http://eprints.iisc.ac.in/id/eprint/53069 View Item