Pseudorandom generators for polynomials

In theoretical computer science, a pseudorandom generator for low-degree polynomials is an efficient procedure that maps a short truly random seed to a longer pseudorandom string in such a way that low-degree polynomials cannot distinguish the output distribution of the generator from the truly random distribution. That is, evaluating any low-degree polynomial at a point determined by the pseudorandom string is statistically close to evaluating the same polynomial at a point that is chosen uniformly at random.

Pseudorandom generators for low-degree polynomials are a particular instance of pseudorandom generators for statistical tests, where the statistical tests considered are evaluations of low-degree polynomials.

Definition
A pseudorandom generator $$G: \mathbb{F}^\ell \rightarrow \mathbb{F}^n$$ for polynomials of degree $$d$$ over a finite field $$\mathbb F$$ is an efficient procedure that maps a sequence of $$\ell$$ field elements to a sequence of $$n$$ field elements such that any $$n$$-variate polynomial over $$\mathbb F$$ of degree $$d$$ is fooled by the output distribution of $$G$$. In other words, for every such polynomial $$p(x_1,\dots,x_n)$$, the statistical distance between the distributions $$p(U_n)$$ and $$p(G(U_\ell))$$ is at most a small $$\epsilon$$, where $$U_k$$ is the uniform distribution over $$\mathbb{F}^k$$.

Construction
The case $$d=1$$ corresponds to pseudorandom generators for linear functions and is solved by small-bias generators. For example, the construction of achieves a seed length of $$\ell= \log n + O(\log (\epsilon^{-1}))$$, which is optimal up to constant factors.

conjectured that the sum of small-bias generators fools low-degree polynomials and were able to prove this under the Gowers inverse conjecture. proved unconditionally that the sum of $$2^d$$ small-bias spaces fools polynomials of degree $$d$$. proves that, in fact, taking the sum of only $$d$$ small-bias generators is sufficient to fool polynomials of degree $$d$$. The analysis of gives a seed length of $$\ell = d \cdot \log n + O(2^d \cdot \log(\epsilon^{-1}))$$.