Abstract
In this paper we study polynomials in VP_{e} (polynomialsized formulas) and in ΣΠΣ (polynomialsize depth3 circuits) whose orbits, under the action of the affine group GL^{aff}_n(𝔽) (the action of (A,b) ∈ GL^{aff}_n(𝔽) on a polynomial f ∈ 𝔽[x] is defined as (A,b)∘f = f(A^Tx+b)), are dense in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms. Specifically, we obtain the following results:
1) For C_n(ℓ_1(x),…,ℓ_n(x)) ≜ Trace(\begin{pmatrix} 𝓁₁(x) & 1 \\ 1 & 0 \end{pmatrix} ⋅ … ⋅ \begin{pmatrix} 𝓁_n(x) & 1 \\ 1 & 0 \end{pmatrix}), where the 𝓁_is are linearly independent linear functions, we construct a polynomialsized interpolating set, and give a polynomialtime reconstruction algorithm. By a result of Bringmann, Ikenmeyer and Zuiddam, the set of all such polynomials is dense in VP_e [Karl Bringmann et al., 2018], thus our construction gives the first polynomialsize interpolating set for a dense subclass of VP_e.
2) For polynomials of the form ANF_Δ(𝓁₁(x),…,𝓁_{4^Δ}(x)), where ANF_Δ(x) is the canonical readonce formula in alternating normal form, of depth 2Δ, and the 𝓁_is are linearly independent linear functions, we provide a quasipolynomialsize interpolating set. We also observe that the reconstruction algorithm of [Ankit Gupta et al., 2014] works for all polynomials in this class. This class is also dense in VP_e.
3) Similarly, we give a quasipolynomialsized hitting set for readonce formulas (not necessarily in alternating normal form) composed with a set of linearly independent linear functions. This gives another dense class in VP_e.
4) We give a quasipolynomialsized hitting set for polynomials of the form f(𝓁₁(x),…,𝓁_{m}(x)), where f is an mvariate ssparse polynomial. and the 𝓁_is are linearly independent linear functions in n ≥ m variables. This class is dense in ΣΠΣ.
5) For polynomials of the form ∑_{i=1}^{s}∏_{j=1}^{d}𝓁_{i,j}(x), where the 𝓁_{i,j}s are linearly independent linear functions, we construct a polynomialsized interpolating set. We also observe that the reconstruction algorithm of [Neeraj Kayal and Chandan Saha, 2019] works for every polynomial in the class. This class is dense in ΣΠΣ. As VP = VNC², 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 kindependent polynomial maps) do not necessarily yield robust hitting sets.
BibTeX  Entry
@InProceedings{medini_et_al:LIPIcs.CCC.2021.19,
author = {Medini, Dori and Shpilka, Amir},
title = {{Hitting Sets and Reconstruction for Dense Orbits in VP\underline\{e\} and \Sigma\Pi\Sigma Circuits}},
booktitle = {36th Computational Complexity Conference (CCC 2021)},
pages = {19:119:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771931},
ISSN = {18688969},
year = {2021},
volume = {200},
editor = {Kabanets, Valentine},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14293},
URN = {urn:nbn:de:0030drops142930},
doi = {10.4230/LIPIcs.CCC.2021.19},
annote = {Keywords: Algebraic complexity, VP, VNP, algebraic circuits, algebraic formula}
}
Keywords: 

Algebraic complexity, VP, VNP, algebraic circuits, algebraic formula 
Collection: 

36th Computational Complexity Conference (CCC 2021) 
Issue Date: 

2021 
Date of publication: 

08.07.2021 