## Abstract

The method of shifted partial derivatives introduced A. Gupta et al. [Approaching the chasm at depth four, IEEE Comp. Soc., 2013, pp. 65-73] and N. Kayal [An exponential lower bound for the sum of powers of bounded degree polynomials, ECCC 19, 2010, p. 81], was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this method alone cannot prove that the padded permanent ℓ^{n-m} perm_{m} cannot be realized inside the GL_{n}^{2} -orbit closure of the determinant detn when n > 2m^{2} +2m. Our proof relies on several simple degenerations of the determinant polynomial, Macaulay's theorem, which gives a lower bound on the growth of an ideal, and a lower bound estimate from [Approaching the chasm at depth four, IEEE Comp. Soc., 2013, pp. 65-73] regarding the shifted partial derivatives of the determinant.

Original language | English |
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Pages (from-to) | 2037-2045 |

Number of pages | 9 |

Journal | Mathematics of Computation |

Volume | 87 |

Issue number | 312 |

DOIs | |

State | Published - 1 Jan 2018 |

## Keywords

- Computational complexity
- Determinant
- Free resolution
- Permanent