## Abstract

In this paper we study polynomials in VPe (polynomial-sized formulas) and in ΣΠΣ (polynomial-size depth-3 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^{T} x + 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. (equation presented), where the ℓis are linearly independent linear functions, we construct a polynomial-sized interpolating set, and give a polynomial-time reconstruction algorithm. By a result of Bringmann, Ikenmeyer and Zuiddam, the set of all such polynomials is dense in VPe [14], thus our construction gives the first polynomial-size interpolating set for a dense subclass of VPe. 2. For polynomials of the form ANF_{∆} (ℓ1(x),..., ℓ_{4}∆ (x)), where ANF_{∆}(x) is the canonical read-once formula in alternating normal form, of depth 2∆, and the ℓis are linearly independent linear functions, we provide a quasipolynomial-size interpolating set. We also observe that the reconstruction algorithm of [35] works for all polynomials in this class. This class is also dense in VPe. 3. Similarly, we give a quasipolynomial-sized hitting set for read-once formulas (not necessarily in alternating normal form) composed with a set of linearly independent linear functions. This gives another dense class in VPe. 4. We give a quasipolynomial-sized hitting set for polynomials of the form f (ℓ1(x),..., ℓm(x)), where f is an m-variate s-sparse polynomial. and the ℓis are linearly independent linear functions in n ≥ m variables. This class is dense in ΣΠΣ. 5. For polynomials of the form ^{Ps}_{i}_{=1}^{Qd}_{j}_{=1} ℓi,j(x), where the ℓi,js are linearly independent linear functions, we construct a polynomial-sized interpolating set. We also observe that the reconstruction algorithm of [45] works for every polynomial in the class. This class is dense in ΣΠΣ. As VP = VNC^{2}, our results for VPe 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.

Original language | English |
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Title of host publication | 36th Computational Complexity Conference, CCC 2021 |

Editors | Valentine Kabanets |

Place of Publication | Dagstuhl, Germany |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 19:1-19:27 |

ISBN (Electronic) | 9783959771931 |

DOIs | |

State | Published - 1 Jul 2021 |

Event | 36th Computational Complexity Conference, CCC 2021 - Virtual, Toronto, Canada Duration: 20 Jul 2021 → 23 Jul 2021 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 200 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 36th Computational Complexity Conference, CCC 2021 |
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Country/Territory | Canada |

City | Virtual, Toronto |

Period | 20/07/21 → 23/07/21 |

## Keywords

- Algebraic circuits
- Algebraic complexity
- Algebraic formula
- VNP
- VP

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