Root Isolation

Root isolation is the process of constructing pairwise disjoint intervals whose endpoints are rational numbers, such that each interval contains exactly one real root of a polynomial and every real root is contained in one of the intervals. Each interval is called an isolating interval.

A squarefree factorization first separates roots of different multiplicities. The intervals can then be found by repeated subdivision using Sturm's theorem, Descartes' sign rule, or continued fraction methods. A bound on root separation provides a sufficient final interval width. For example, the two real roots of are isolated by

An alternative exact representation is a Thom encoding, which identifies a real root by the signs of the successive derivatives of its polynomial.