Arkadev Chattopadhyay — Page 2

arXiv2013

Factoring bivariate lacunary polynomials without heights

We present an algorithm which computes the multilinear factors of bivariate lacunary polynomials. It is based on a new Gap Theorem which allows to test whether a polynomial of the form P(X,X+1) is identically zero in time polynomial in the number of terms of P(X,Y). The algorithm we obtain is more elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on the valuation of polynomials of the previous form instead of the height of the coefficients. As a result, it can be used to find some linear factors of bivariate lacunary polynomials over a field of large finite characteristic in probabilistic polynomial time.

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arXiv2008

Multiparty Communication Complexity of Disjointness

We obtain a lower bound of n^Omega(1) on the k-party randomized communication complexity of the Disjointness function in the `Number on the Forehead' model of multiparty communication when k is a constant. For k=o(loglog n), the bounds remain super-polylogarithmic i.e. (log n)^omega(1). The previous best lower bound for three players until recently was Omega(log n). Our bound separates the communication complexity classes NP^{CC}_k and BPP^{CC}_k for k=o(loglog n). Furthermore, by the results of Beame, Pitassi and Segerlind \cite{BPS07}, our bound implies proof size lower bounds for tree-like, degree k-1 threshold systems and superpolynomial size lower bounds for Lovasz-Schrijver proofs. Sherstov \cite{She07b} recently developed a novel technique to obtain lower bounds on two-party communication using the approximate polynomial degree of boolean functions. We obtain our results by extending his technique to the multi-party setting using ideas from Chattopadhyay \cite{Cha07}. A similar bound for Disjointness has been recently and independently obtained by Lee and Shraibman.

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