Noncommutativity Makes Determinants Hard

Markus Bläser

Abstract: We consider the complexity of computing the determinant over arbitrary finite-dimensional algebras. We first consider the case that A is fixed. We obtain the following dichotomy: If A/rad A is noncommutative, then computing the determinant over A is hard. “Hard” here means #P-hard over fields of characteristic 0 and Mod p P-hard over fields of characteristic p > 0. If A/ rad A is commutative and the underlying field is perfect, then we can compute the determinant over A in polynomial time.