The Distribution of Condition Numbers of Rational Data of Bounded Bit Length

We prove that rational data of bounded input length are uniformly distributed (in the classical sense of H. Weyl, in [42]) with respect to the probability distribution of condition numbers of Numerical Analysis. We deal both with condition numbers of Linear Algebra and with condition numbers for systems of multivariate polynomial equations. For instance, we prove that for a randomly chosen n×n rational matrix M of bit length O(n log n) + log w, the condition number k(M) satisfies k(M) ≤ wn with probability at least 1 − 2w−1. Similar estimates are established for the condition number μnorm of M. Shub and S. Smale when applied to systems of multivariate homogeneous polynomial equations of bounded input length. Finally, we apply these techniques to estimate the probability distribution of the precision (number of bits of the denominator) required to write down approximate zeros of systems of multivariate polynomial equations of bounded input length.