Pseudorandom generators (PRGs) for low-degree polynomials are a central object in pseudorandomness, with applications to circuit lower bounds and derandomization. Viola's celebrated construction gives a PRG over the binary field, but with seed length exponential in the degree $d$. This exponential dependence can be avoided over sufficiently large fields. In particular, Dwivedi, Guo, and Volk constructed PRGs with optimal seed length over fields of size exponential in $d$. The latter builds on the framework of Derksen and Viola, who obtained optimal-seed constructions over fields of size polynomial in $d$, although growing with the number of variables $n$. In this work, we construct the first PRG with optimal seed length for degree-$d$ polynomials over fields of polynomial size, specifically $q \approx d^4$, assuming sufficiently large characteristic. Our construction follows the framework of prior work and reduces the required field size by replacing the hitting-set generator used in previous constructions with a new pseudorandom object. We also observe a threshold phenomenon in the field-size dependence. Specifically, we prove that constructing PRGs over fields of sublinear size, for example $q = d^{0.99}$ where $q$ is a power of two, would already yield PRGs for the binary field with comparable seed length via our reduction, provided that the construction imposes no restriction on the characteristic. While a breakdown of existing techniques has been noted before, we prove that this phenomenon is inherent to the problem itself, irrespective of the technique used.

翻译：低次多项式的伪随机生成器是伪随机性理论的核心研究对象，在电路下界和去随机化领域具有重要应用。Viola的著名构造给出了二元域上的伪随机生成器，但其种子长度随次数$d$呈指数增长。在足够大的域上可以避免这种指数依赖关系。特别地，Dwivedi、Guo和Volk在规模为$d$的指数函数的域上构造了具有最优种子长度的伪随机生成器。后者的工作建立在Derksen和Viola的理论框架之上，他们在规模为$d$的多项式函数（尽管随变量数$n$增长）的域上获得了最优种子构造。在本研究中，我们首次构造了在多项式规模域（具体而言$q \approx d^4$）上针对$d$次多项式的最优种子长度伪随机生成器，该构造假设域特征足够大。我们的构造遵循先前工作的理论框架，通过用新的伪随机对象替代先前构造中使用的命中集生成器，从而降低了所需域规模。我们还观察到域规模依赖关系中的阈值现象。具体而言，我们证明：若构造不限制域特征，在亚线性规模域（例如$q = d^{0.99}$，其中$q$为2的幂）上构造伪随机生成器，通过我们的归约方法将能产生具有可比种子长度的二元域伪随机生成器。尽管现有技术存在的局限性已被指出，但我们证明该现象是问题本身固有的特性，与所采用的技术无关。