A Near-Optimal Expected Boolean Complexity Bound for Descartes Solver

Isolating the real roots of univariate polynomials is a fundamental problem in symbolic computation and it is arguably one of the most important problems in computational mathematics. The problem has a long history decorated with numerous ingenious algorithms and furnishes an active area of research. However, the worst-case analysis of root-finding algorithms does not correlate with their practical performance. We develop a smoothed analysis framework on polynomials with integer coefficients to bridge the gap between the complexity estimates and the practical performance. In particular, we show that the Descartes solver has near-optimal expected Boolean complexity.