Suppose you're using a symbolic algebra system & want to estimate the rank of a matrix whose entries are symbolic expressions involving many different variables. My intuition tells me that I should be able to randomly guess values for all of the variables, numerically compute the rank for that assignment of variables, and do it again several more times. The *maximum* rank achieved should be close to the actual symbolic rank of the matrix. 2 cases: integers and floating point. If the symbolic expressions enable exact integer values, then the rank could be computed using arbitrary-precision arithmetic. However, it might be more efficient to do the computation modulo a large number of different primes. If we work in floating point, the expressions could be incredibly ill-conditioned, in which case telling the difference between very small numbers and actual zeros could be impossible. Other than such ill-conditioning, is it possible that the rank could be *over* estimated by such random sampling?