Probabilistic Condition Number Estimates For Real Polynomial Systems I: A Broader Family Of Distributions
This work provides theoretical guarantees for the sensitivity of polynomial systems, benefiting numerical analysts and researchers in computational mathematics.
The paper establishes new probabilistic estimates for the condition number of real polynomial systems, allowing a broader family of measures and over-determined systems, and derives new Lipschitz estimates for polynomial maps.
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, 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. Along the way, we derive new Lipshitz estimates for polynomial maps from R^n to R^m, extending earlier work of Kellog on the case m=1, which may be of independent interest. In Part II, we study smoothed complexity and how sparsity (in the sense of restricting which monomial terms can appear) can help further improve earlier condition number estimates.