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This paper was removed due to an error in the proof (Claim 4.12 as stated is not true). The authors would like to thank Ilya Volkovich for pointing out a counterexample to this papers main result in positive characteristic: If $F$ is a field with prime characteristic $p$, then the polynomial $x_1^p + x_2^p + ldots + x^n^p$ has the following factor: $(x_1+x_2+ ldots + x_n)^{p-1}$, which has sparsity $n^p$.
Random subspaces $X$ of $mathbb{R}^n$ of dimension proportional to $n$ are, with high probability, well-spread with respect to the $ell_p$-norm (for $p in [1,2]$). Namely, every nonzero $x in X$ is robustly non-sparse in the following sense: $x$ is $
In analogy with the regularity lemma of Szemeredi, regularity lemmas for polynomials shown by Green and Tao (Contrib. Discrete Math. 2009) and by Kaufman and Lovett (FOCS 2008) modify a given collection of polynomials calF = {P_1,...,P_m} to a new co
Mahaneys Theorem states that, assuming $mathsf{P} eq mathsf{NP}$, no NP-hard set can have a polynomially bounded number of yes-instances at each input length. We give an exposition of a very simple unpublished proof of Manindra Agrawal whose ideas a
We determine the rank of a random matrix over an arbitrary field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula vindicates a conje
A star $k$-coloring is a proper $k$-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the