### Abstract

The Descartes test based on Bernstein coefficients and Descartes’ rule of signs effectively (over-)approximates the number of real roots of a univariate polynomial over an interval. In this entry we formalise the theorem of three circles, which gives sufficient conditions for when the Descartes test returns 0 or 1. This is the first step for efficient root isolation.

### License

BSD License

### Topics

- Mathematics/Analysis

### Session Three_Circles

- RRI_Misc
- Bernstein_01
- Bernstein
- Normal_Poly
- Three_Circles

### Depends on

- The Budan–Fourier Theorem and Counting Real Roots with Multiplicity
- Polynomial Interpolation

### Auto-related entries

- Symmetric Polynomials

## Cite

×

Three_Circles-AFP

    @article{Three_Circles-AFP,
      author  = {Fox Thomson and Wenda Li},
      title   = {The Theorem of Three Circles},
      journal = {Archive of Formal Proofs},
      month   = {August},
      year    = {2021},
      note    = {\url{https://isa-afp.org/entries/Three_Circles.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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Older releases:

- Sep 13, 2023 : Isabelle2023
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- Dec 14, 2021 : Isabelle2021-1
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