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Quasi-random multilinear polynomials

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Abstract

We consider multilinear Littlewood polynomials, polynomials in n variables in which a specified set of monomials U have ±1 coefficients, and all other coefficients are 0. We provide upper and lower bounds (which are close for U of degree below log n) on the minimum, over polynomials h consistent with U, of the maximum of |h| over ±1 assignments to the variables. (This is a variant of a question posed by Erdős regarding the maximum on the unit disk of univariate polynomials of given degree with unit coefficients.) We outline connections to the theory of quasi-random graphs and hypergraphs, and to statistical mechanics models. Our methods rely on the analysis of the Gale–Berlekamp game; on the constructive side of the generic chaining method; on a Khintchine-type inequality for polynomials of degree greater than 1; and on Bernstein’s approximation theory inequality.

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel

    Gil Kalai

  2. Division of Engineering and Applied Science, California Institute of Technology, MC305-16, Pasadena, CA, 91125, USA

    Leonard J. Schulman

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  1. Gil Kalai
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  2. Leonard J. Schulman
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Correspondence to Leonard J. Schulman.

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Kalai, G., Schulman, L.J. Quasi-random multilinear polynomials. Isr. J. Math. 230, 195–211 (2019). https://doi.org/10.1007/s11856-018-1821-y

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  • DOI: https://doi.org/10.1007/s11856-018-1821-y

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