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    Challenges in computational lower bounds

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    We draw two incomplete, biased maps of challenges in computational complexity lower bounds

    Constructive Relationships Between Algebraic Thickness and Normality

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    We study the relationship between two measures of Boolean functions; \emph{algebraic thickness} and \emph{normality}. For a function ff, the algebraic thickness is a variant of the \emph{sparsity}, the number of nonzero coefficients in the unique GF(2) polynomial representing ff, and the normality is the largest dimension of an affine subspace on which ff is constant. We show that for 0<ϵ<20 < \epsilon<2, any function with algebraic thickness n3ϵn^{3-\epsilon} is constant on some affine subspace of dimension Ω(nϵ2)\Omega\left(n^{\frac{\epsilon}{2}}\right). Furthermore, we give an algorithm for finding such a subspace. We show that this is at most a factor of Θ(n)\Theta(\sqrt{n}) from the best guaranteed, and when restricted to the technique used, is at most a factor of Θ(logn)\Theta(\sqrt{\log n}) from the best guaranteed. We also show that a concrete function, majority, has algebraic thickness Ω(2n1/6)\Omega\left(2^{n^{1/6}}\right).Comment: Final version published in FCT'201
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