### Abstract

Based on evaluating Cauchy indices through remainder sequences, this
entry provides an effective procedure to count the number of complex
roots (with multiplicity) of a polynomial within various shapes (e.g., rectangle,
circle and half-plane). Potential applications of this entry include certified
complex root isolation (of a polynomial) and testing the Routh-Hurwitz
stability criterion (i.e., to check whether all the roots of some
characteristic polynomial have negative real parts).

### License

### History

- October 26, 2021
- resolved the roots-on-the-border problem in the rectangular case (revision 82a159e398cf).