Hitting Sets and Reconstruction for Dense Orbits in VP_e and ΣΠΣ Circuits
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In this paper we study polynomials in VP_e (polynomial-sized formulas) and in ΣΠΣ (polynomial-size depth-3 circuits) whose orbits, under the action of the affine group GL_n^aff(𝔽), are 𝑑𝑒𝑛𝑠𝑒 in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms. As VP=VNC^2, our results for VP_e translate immediately to VP with a quasipolynomial blow up in parameters. If any of our hitting or interpolating sets could be made 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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