A sequence of polynomials obtained by applying the Euclidean algorithm to a real univariate polynomial and its derivative (or using a signed remainder sequence) whose sign variations evaluated at interval endpoints count the number of distinct real roots inside that interval (Sturm's theorem).

Principle

Construct the remainder sequence p0=p, p1=p', p2=−rem(p0,p1), p3=−rem(p1,p2), … until termination; evaluate the sequence at endpoints a and b and compute the difference in the number of sign changes to obtain the count of distinct real roots of p in (a,b).

Demonstration

For p(x)=x^3−x, compute p and p', form the signed remainder sequence, evaluate at a=−2 and b=2; the difference in sign variations gives the count of real roots in (−2,2), and isolating intervals can be refined by bisection with Sturm counts.

Misapplication

Using Sturm sequences without handling repeated roots (multiplicities cause zero entries and require deflation) or evaluating with floating‑point values without guaranteed sign determination can produce incorrect root counts; miscomputing signed remainders breaks the theorem's conditions.

Consequence

A correct Sturm sequence gives an exact integer count of distinct real roots in an interval and supports certified root isolation and multiplicity‑aware refinement procedures useful in exact real algebraic computation and CAD projection steps.

Reversal

Instead of Sturm sequences one might use Descartes' Rule of Signs or Budan‑Fourier bounds which provide upper bounds on positive roots or counts modulo multiplicity but do not in general yield exact counts per interval without additional refinements.

Boundary

Applies to univariate polynomials with real coefficients and uses Euclidean division in fields of characteristic zero or with suitable adaptations; it does not directly generalize to multivariate polynomials without reduction to univariate cases or to polynomials over finite fields without corresponding analogs.

Semantic Tension

Sturm sequences provide exact root counts by sign variation while subresultant PRS methods and modern numeric root‑finding trade symbolic exactness for computational efficiency; there is tension between exact certification and performance in large or approximate problems.

Synthesis

A Sturm Sequence is a signed Euclidean remainder chain between a real polynomial and its derivative whose sign variation at interval endpoints yields the exact number of distinct real roots inside the interval; properly implemented it is a foundational certified tool for real root counting and isolation.