On the uniform distribution of rational inputs with respect to condition numbers of Numerical Analysis

We show that rational data of bounded input length are uniformly distributed with respect to 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 show that for any $w>1$ and for any $n\times n$ rational matrix $M$ of bit length $O(n^4\log n) + \log w$, the condition number $k(M)$ satisfies $k(M) \leq w n^{5/2}$ with probability at least $1-2w^{-1}$. Similar estimates are shown for the condition number $\mu_{norm}$ 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 show the probability distribution of the precision (number of bits of the denominator) required to write down approximate zeros of affine systems of multivariate polynomial equations of bounded input length.