A note on the complexity of univariate root isolation

This paper presents the average-case bit complexity of subdivision-based univariate solvers, namely those named after Sturm, Descartes, and Bernstein. By real solving we mean real root isolation. We prove bounds of $\sOB(N^5)$ for all methods, where $N$ bounds the polynomial degree and the coefficient bitsize, whereas their worst-case complexity is in $\sOB(N^6)$. In the case of the Sturm solver, our bound depends on the number of real roots. Our work is a step towards understanding the practical complexity of real root isolation. This enables a better juxtaposition against numerical solvers, the latter having worst-case complexity in $\sOB(N^4)$. % Our approach extends to complex root isolation, where we offer a simple proof leading to bounds % for the number of steps that the subdivision algorithm performs on the worst and average-case complexities of $\sOB(N^7 )$ and $\sOB(N^6)$ respectively, where the latter is output-sensitive.