The Budan-Fourier Theorem gives an upper bound on the number of real roots (counting multiplicity) of a real polynomial in a given interval $I=(a,b]$, where $a=-\infty$ and/or $b=\infty$ are allowed. The algorithm is simple. Take a (nonzero) polynomial and compute all of its derivatives. Evaluate this list of derivatives at $x=a$ and create a list of results. After ignoring all zeros, count the number of times the list of numbers switches from positive to negative numbers or vice-versa (we include "$\pm\infty$" as possible "numbers"). Next, do the same for $x=b$.
Budan and Fourier showed that the number of sign changes at $x=a$ minus the number of sign changes at $x=b$ provides an upper bound on the number of real roots of that polynomial (counting multiplicity). In fact, if the difference in sign changes is $s$, then this polynomial has $s-2k$ real roots (counting multiplicity) for some positive integer $k$.
For example, if $s=3$, then there are either 3 or 1 real roots. On the other hand, if $s=4$, then there are either 4, 2, or no real roots.
It is interesting to note that Descartes' Rule of Signs is a special case of this result. If we set $a=0$, the sign changes in $f(0),f'(0),f''(0),\dots$ is the same as the sign changes in the coefficients of $f(x)$. If $b=\infty$, then all of signs of $f(\infty),f'(\infty),f''(\infty),\dots$ will be the same. Thus Descartes' Rule of Signs counts the same thing as Budan-Fourier for $a=0$ and $b=\infty$: the roots in $I=(0,\infty)$ (i.e. the positive real roots).
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