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.
Presumably by "four axioms" you refer to the properties that define a polynomial Sturm sequence. Sturm's polynomial root-counting algorithm algorithm works in any real-closed-field. Recall that a field is real if $\,-1\,$ is not a sum of squares, and a real-closed field is a real field with no proper real algebraic extension. There are many equivalent ways to characterize real-closed fields $\,R,\,$ e.g. either $\,r\,$ or $\,-r\,$ has a square-root $\in R$ and every odd-degree polynomial $\ f(x)\in R[x]$ has a root $\in R.\,$ Alternatively: $\ R\ $ is an ordered field wrt the order $\ r\ge 0\iff r\ $ is a square, and $\ R\ $ has the intermediate value property wrt this order.
Sturm's algorithm plays a fundamental role in the theory of real-closed fields. For example, it easily yields the uniqueness of a real-closure fixing a given order. Further it can be generalized to prove that the projection of a semi-algebraic set is semi-algebraic - a geometric form of Tarski's celebrated quantifier elimination for real-closed fields. This leads to an effective algorithm for deciding the truth of first order statements in real-closed fields - by cylindric algebraic decomposition (CAD). The algorithm simply decomposes $\, R^n\,$ into a finite number of constant-sign "cylinders" and then simply inspects the sign on each component (this is completely obvious in the one-dimensional case, e.g. see here). You can read about this in any textbook on real algebraic geometry.