We prove that the sum of d small-bias generators L : F{sup}s → F{sup}n fools degree-d polynomials in n variables over a prime field F, for any fixed degree d and field F, including F = F{sub}2 = {0, 1}. Our result improves on both the work by Bogdanov and Viola (FOCS '07) and the beautiful follow-up by Lovett (STOC '08). The first relies on a conjecture that turned out to be true only for some degrees and fields, while the latter considers the sum of 2{sup}d small-bias generators (as opposed to d in our result). Our proof builds on and somewhat simplifies the arguments by Bogdanov and Viola (FOCS '07) and by Lovett (STOC '08). Its core is a case analysis based on the bias of the polynomial to be fooled.
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机译：我们证明了D小偏置发生器L：f {sup} s→f {sup} n over在素材字段f上的n个变量中的D度-D多项式，适用于任何固定等级的d和字段f，包括f = f {sub} 2 = {0,1}。我们的结果有关Bogdanov和Viola的工作（Focs'07）和Lovett的美丽随访（STOC '08）。第一个依赖于猜想，该猜想仅为某些程度和字段而真实，而后者考虑了2 {SUP} D小偏置发生器的总和（而不是我们的结果）。我们的证据构建，有点简化了Bogdanov和Viola（Focs'07）和Lovett（STOC'08）的论据。它的核心是基于多项式的偏见被愚弄的案例分析。
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