Speaker: |
Mrinal Kumar (Simons Institute for the Theory of Computing Berkeley, CA 94720-2190) |

Organiser: |
Ramprasad Saptharishi |

Date: |
Monday, 7 Jan 2019, 11:00 to 12:00 |

Venue: |
A-201 (STCS Seminar Room) |

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A fundamental question in this line of research, which has largely remained open is to understand if other natural classes of multivariate polynomials, for instance, arithmetic formulas, algebraic branching programs, constant depth arithmetic circuits or the complexity class VNP (the algebraic analog of NP) of polynomials, are closed under taking factors. In addition to being fundamental questions on their own, such 'closure results' for polynomial factorization play a crucial role in the understanding of hardness randomness tradeoffs for algebraic computation.

I will talk about the following two results, whose study was motivated by these questions.

1. The class VNP is closed under taking factors.

This proves a conjecture of Bürgisser.

2. All factors of degree at most poly(log n) of polynomials with constant depth circuits of size

poly(n) have constant (a slightly larger constant) depth arithmetic circuits of size poly(n).

This partially answers a question of Shpilka and Yehudayoff and has applications to hardness-randomness tradeoffs for constant depth arithmetic circuits.

Based on joint work with Chi-Ning Chou and Noam Solomon.