In this paper we study polynomials in $\text{VP}_e$ (polynomial-sized formulas) and in $\Sigma\Pi\Sigma$ (polynomial-size depth-$3$ circuits) whose orbits, under the action of the affine group $\text{GL}_n^{\text{aff}}(\mathbb{F})$, are $\mathit{dense}$ in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms. As $\text{VP}=\text{VNC}^2$, our results for $\text{VP}_e$ translate immediately to $\text{VP}$ with a quasipolynomial blow up in parameters. If any of our hitting or interpolating sets could be made $\mathit{robust}$ then this would immediately yield a hitting set for the superclass in which the relevant class is dense, and as a consequence also a lower bound for the superclass. Unfortunately, we also prove that the kind of constructions that we have found (which are defined in terms of $k$-independent polynomial maps) do not necessarily yield robust hitting sets.
翻译:本文用$\ text{VP}}(\ mathbb{F}) 和 $Sigma\ Pi}SigmaIal用$( Pi\SigmaIal 尺寸的公式) 和 $\Sgimoimal 深度- $3美元电路) 进行多数值研究, 其轨道在finde小组 $\ text{GL ⁇ n}}(\mathbb{F}) 的操作下, 在其环境等级下, $\ mathit{dense} (\ mathatit{robust} $ 。 我们为这些轨道建造了撞击器和内插接合器, 并提供了重建算法。 正如 $\ text{Vp* text{VNC} 2$ 一样, 我们的 $ 的计算结果立即翻译为$\ text{VPe $ 和 参数的准极值。 如果我们的任何打击或猜测合装置都能确定我们所找到的多值。