# On the Hardness of the Noncommutative Determinant

Srikanth Srinivasan
(Institute of Mathematical Sciences, Chennai)
Beregningsmatematikseminar
Tirsdag, 23 februar, 2010, at 14:15-16:00, Turing-014 at Department of Computer Science
Abstrakt:

We study the computational complexity of computing the noncommutative determinant. We first consider the arithmetic circuit complexity of computing the noncommutative determinant polynomial. Then, more generally, we also examine the complexity of computing the determinant (as a function) over noncommutative domains. Our hardness results are summarized below:

1. We show that if the noncommutative determinant polynomial has small noncommutative arithmetic circuits then so does the noncommutative permanent, which would imply that the commutative permanent polynomial would have small commutative arithmetic circuits.
2. For any field F we show that computing the n X n permanent over F is polynomial-time reducible to computing the 2n X 2n (noncommutative) determinant whose entries are O(n^2) X O(n^2) matrices over the field F.
3. We also derive as a consequence that computing the n X n permanent over nonnegative rationals is polynomial-time reducible to computing the noncommutative determinant over Clifford algebras of n^{O(1)} dimension.

Our techniques are elementary and use primarily the notion of the Hadamard Product of noncommutative polynomials.

Preprint: arXiv:0910.2370

Host: Peter Bro Miltersen

Kontaktperson: Peter Bro Miltersen