As we all know, Sturm's axioms have completely solved the problem for finding the number of roots in an arbitrary interval $[a,b]$, using the derivative and forms a Sturm set.
Now my question follows naturally that is there any other use of this axiom.
When I learned the Sturm set, I thought it will be difficult to find any such set for an arbitrary polynomial, nonetheless, the use of derivatives is really marvelous, and from then on, this question has occupied a part of my mind, shouting at me, asking me to treat it seriously.
Any response is well appreciated, thanks.
Edit: The four axioms are as follows as far as I am concerned:
Given a set of polynomials ordered by natural numbers and the 0-th is the desired polynomial.
(1): $\forall i \in \{1, \ldots, n\}$, $f_{i}$ and $f_{i+1}$ do not share the same root.
(2):$f_n(x)$ does not have one root.
(3):If $\mathfrak{a}$ is a root of $f_k$, then $f_{k-1}(\mathfrak{a})$ and $f_{k+1}(\mathfrak{a})$ do not have the same sign.
(4):If $\mathfrak{b}$ is a root of f(x), then in the interval $(-\infty, \mathfrak{b})$, $f_0(\mathfrak{b})$ and $f_1(\mathfrak{b})$ do not have the same sign; and in the interval $(\mathfrak{b},\infty)$, they share the same sign.
After reading the answer by @Bill Dubuque, since I am not familiar with the theory of algorithms, I found the Sturm's set to be full of unfamiliar things. In any case, thanks very much.