### Abstract

Two polynomials f, g ∈ double-struck F[x_{1}...,x_{n}] are called shift-equivalent if there exists a vector (a_{1}..., a _{n} ∈ double-struck F^{n} such that the polynomial identity f(x_{1}+a_{1},...,x_{n}+a_{n}) ≡g(x_{1},...,x_{n}) holds. Our main result is a new randomized algorithm that tests whether two given polynomials are shift equivalent. Our algorithm runs in time polynomial in the circuit size of the polynomials, to which it is given black box access. This complements a previous work of Grigoriev [Gri97 who gave a deterministic algorithm running in time n^{O(d)} for degree d polynomials. Our algorithm uses randomness only to solve instances of the Polynomial Identity Testing (PIT) problem. Hence, if one could de-randomize PIT (a long-standing open problem in complexity) a de-randomization of our algorithm would follow. This establishes an equivalence between de-randomizing shift-equivalence testing and de-randomizing PIT (both in the black-box and the white-box setting). For certain restricted models, such as Read Once Branching Programs, we already obtain a deterministic algorithm using existing PIT results.

Original language | English (US) |
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Title of host publication | Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings |

Publisher | Springer Verlag |

Pages | 417-428 |

Number of pages | 12 |

Edition | PART 1 |

ISBN (Print) | 9783662439470 |

DOIs | |

State | Published - Jan 1 2014 |

Event | 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 - Copenhagen, Denmark Duration: Jul 8 2014 → Jul 11 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 1 |

Volume | 8572 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 |
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Country | Denmark |

City | Copenhagen |

Period | 7/8/14 → 7/11/14 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings*(PART 1 ed., pp. 417-428). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8572 LNCS, No. PART 1). Springer Verlag. https://doi.org/10.1007/978-3-662-43948-7_35