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\title{Count the Number of Complex Roots}
\author{Wenda Li}
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\begin{abstract}
  Based on evaluating Cauchy indices through remainder sequences \cite{eisermann2012fundamental} \cite[Chapter 11]{rahman2002analytic}, this entry provides an effective procedure to count the number of complex roots (with multiplicity) of a polynomial within a rectangle box or a 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).
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\section{Acknowledgements}
The work was supported by the ERC Advanced Grant ALEXANDRIA (Project 742178), funded by the European Research Council
and led by Professor Lawrence Paulson at the University of Cambridge, UK.

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