On the enumerability of the determinant and the rank

We investigate the complexity of enumerative approximation of two elementary problems in linear algebra, computing the rank and the determinant of a matrix. In particular, we show that if there exists an enumerator that, given a matrix, outputs a list of constantly many numbers, one of which is guaranteed to be the rank of the matrix, then it can be determined in AC⁰ (with oracle access to the enumerator) which of these numbers is the rank. Thus, for example, if the enumerator is an FL function, then the problem of computing the rank is in FL. The result holds for matrices over any commutative ring whose size grows at most polynomially with the size of the matrix. The existence of such an enumerator also implies a slightly stronger collapse of the exact counting logspace hierarchy. For the determinant function we establish the following two results: (1) If the determinant is poly-enumerable in logspace, then it can be computed exactly in FL. (2) For any prime p, if computing the determinant modulo p is (p - 1)-enumerable in FL, then computing the determinant modulo p can be done in FL. This gives a new perspective on the approximability of many elementary linear algebra problems equivalent to computing the rank or the determinant.