We consider the problem of constructing hitting set generators for sparse multivariate polynomials over any finite fields. Hitting set generators, just as pseudorandom generators, play a fundamental role in the study of derandomization. Pseudorandom generators with a logarithmic seed length are only known for polynomials of a constant degree cite{Lov09, Vio09}. On the other hand, hitting set generators with a logarithmic seed length are known for polynomials of larger degrees, but only over fields which are much larger than the degrees cite{KS01, Bog05}. Our main result is the construction of a hitting set generator with a seed length of $O(log s)$, which works for $s$-term polynomials of any degrees over any finite fields of constant size. This gives the first optimal hitting set generator which allows the fields to be smaller than the degrees of polynomials. For larger fields, of non-constant size, we provide another hitting set generator with a seed length of $O(log (sd))$, which works for $s$-term polynomials of any degree $d$, as long as $d$ is slightly smaller than the field size.
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机译:我们考虑为任何有限域上的稀疏多元多项式构造命中集生成器的问题。就像伪随机生成器一样,命中集生成器在去随机化研究中也起着基本作用。具有对数种子长度的伪随机生成器仅对于恒定度数的多项式为已知{Lov09,Vio09}。另一方面,已知具有对数种子长度的命中集合生成器用于较大次数的多项式,但是仅在比引用次数大得多的字段上{KS01,Bog05}。我们的主要结果是构造了种子长度为$ O(log s)$的打击集生成器,该生成器可用于在恒定大小的任何有限域上任意度数的$ s $项多项式。这给出了第一个最佳的命中集生成器,它使场小于多项式的次数。对于非恒定大小的较大字段,我们提供了另一个命中集生成器,其种子长度为$ O(log(sd))$,它可用于$ s $次任意多项式$ d $的项,只要$ d $略小于字段大小。
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