Hitting Sets and Reconstruction for Dense Orbits in $\text{VP}_e$ and $ΣΠΣ$ Circuits

In this paper we study polynomials in $\text{VP}_e$ (polynomial-sized formulas) and in $ΣΠΣ$ (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.

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