### 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

BSD License

### History

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

### Topics

- Mathematics/Analysis

### Session Count_Complex_Roots

- CC_Polynomials_Extra
- Extended_Sturm
- Count_Line
- Count_Half_Plane
- Count_Circle
- Count_Rectangle
- Count_Complex_Roots
- Count_Complex_Roots_Examples

### Depends on

- Polynomial Interpolation
- The Sturm–Tarski Theorem
- Evaluate Winding Numbers through Cauchy Indices

### Used by

- Linear Recurrences

## Cite

×

Count_Complex_Roots-AFP

    @article{Count_Complex_Roots-AFP,
      author  = {Wenda Li},
      title   = {Count the Number of Complex Roots},
      journal = {Archive of Formal Proofs},
      month   = {October},
      year    = {2017},
      note    = {\url{https://isa-afp.org/entries/Count_Complex_Roots.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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Older releases:

- Sep 13, 2023 : Isabelle2023
- Oct 27, 2022 : Isabelle2022
- Dec 14, 2021 : Isabelle2021-1
- Feb 23, 2021 : Isabelle2021
- Apr 18, 2020 : Isabelle2020
- Jun 11, 2019 : Isabelle2019
- Aug 16, 2018 : Isabelle2018
- Oct 18, 2017 : Isabelle2017