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  1. ; ; (, Foundations of Computational Mathematics)
    We consider the sensitivity of real roots of polynomial systems with respect to perturbations of the coefficients. In particular --- for a version of the condition number defined by Cucker and used later by Cucker, Krick, Malajovich, and Wschebor --- we establish new probabilistic estimates that allow a much broader family of measures than considered earlier. We also generalize further by allowing over-determined systems. In Part II, we study smoothed complexity and how sparsity (in the sense of restricting which terms can appear) can help further improve earlier condition number estimates. 
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