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### Abstract

This entry is mainly about counting and approximating real roots (of a
polynomial) with multiplicity. We have first formalised the
Budan-Fourier theorem: given a polynomial with real coefficients, we
can calculate sign variations on Fourier sequences to over-approximate
the number of real roots (counting multiplicity) within an interval.
When all roots are known to be real, the over-approximation becomes
tight: we can utilise this theorem to count real roots exactly. It is
also worth noting that Descartes' rule of sign is a direct
consequence of the Budan-Fourier theorem, and has been included in
this entry. In addition, we have extended previous formalised
Sturm's theorem to count real roots with multiplicity, while the
original Sturm's theorem only counts distinct real roots.
Compared to the Budan-Fourier theorem, our extended Sturm's
theorem always counts roots exactly but may suffer from greater
computational cost.