Jacques Sturm developed a method that computes the number of distinct real roots of a real polynomial. This technique can be used to count the total number of real roots or just count the number of roots lying in a particular interval.
Sturm says to take a polynomial $r_0=p(x)$ and compute its derivative $r_1=p'(x)$. Then divide $p(x)$ by $p'(x)$. This yields a polynomial quotient and remainder. Let $r_2=-\mathrm{Remainder}(r_0,r_1)$ be the negative of the remainder. Now continue dividing $r_{i-1}$ by $r_i$ setting $r_{i+1}$ equal to the negative remainder until a zero remainder is achieved (this must happen eventually since each division yields a negative remainder of lower degree).
Next, take the list negative remainders (as well as $p(x)$ and $p'(x)$) and plug in $x=a$. This yields a list of real numbers. Count the number of sign changes in this list (ignoring zeros) and call the number of sign changes $A$. Then do the same for $x=b$ and call the number of sign changes $B$. Then if neither $x=a$ nor $x=b$ is a repeated root, the number of distinct real roots in the interval $I=(a,b] = \{ x \in \mathbb{R} \;|\; a \lt x \leq b \}$ is exactly $A-B$.
When either $x=a$ or $x=b$ is a repeated root, substituting that number into the sequence of polynomials will yield a string of 0's. But all is not lost, if the entire sequence is divided by its last nonzero term, the modified Sturm sequence then applies as a method. Since the modified Sturm sequence works regardless of our choice of $x=a$ and $x=b$, this demo always uses the modified sequence.
Note that we can "evaluate the Sturm Sequence at $\pm\infty$" (where limits are actually being taken to $\pm\infty$) and use this method to find the total number of distinct real roots of $p(x)$ on the whole real line $\mathbb{R}=(-\infty,\infty)$.
Sturm's method can be extended to count real roots with multiplicity. The idea is that $p_1(x)=\mathrm{gcd}(p(x),p'(x))$ contains any repeated copies of factors of $p(x)$. If Sturm's method is run on $p_1(x)$, the number of at least doubly repeated real roots will be reported. If Sturm's method is then run on $p_2(x)=\mathrm{gcd}(p_1(x),p_1'(x))$, the number of at least triply real roots will be reported. This iterative process will eventually terminate and reveal the number real roots of various multiplicities.
The Sage demo below uses Sturm's method (applied to modified Sturm sequences) to compute the number of real roots of various multiplicities. In addition, the demo gives an option to display a table of sign patterns for the first Sturm sequence which computes the number of distinct real roots.
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