Sturm Distinct Real Root Count
Certify how many distinct real roots lie in an interval, including repeated-root and endpoint cases.
Free theory articleCertify how many distinct real roots lie in an interval, including repeated-root and endpoint cases.
Concept & interpretation
Sturm's theorem provides a powerful method to determine the exact number of distinct real roots of a polynomial within a given interval. Unlike numerical root-finding methods that approximate solutions, Sturm's theorem offers a mathematical certainty—a certified count—by analyzing the sign variations in a sequence of polynomials derived from the original function.
Experiment with Polynomials
You can use the interactive module to explore different polynomials and intervals: https://riano.app/module/sturm-distinct-root-count.
- Select the Repeated polynomial: x3 - 3x + 2.
- Set Endpoint a to -2.
- Set Endpoint b to 1.
- Observe the certified count: Distinct roots in interval is 1. The square-free degree is shown as 2.
- Change the polynomial to Distinct (x3 - x) and the interval to [-1, 0]. The count updates to reflect the roots in that specific region.