Certified root counting with Hermite matrices
Count the real roots of a polynomial from the signature of its Hermite matrix, and compare with Sturm sequences.
- Suitable for
- Undergraduate thesis
- Supervision
- Tülay Ayyıldız
- Prerequisites
- Linear algebra; programming in Julia or Python
The Hermite quadratic form of a univariate polynomial encodes its roots: its rank gives the number of distinct complex roots and its signature the number of distinct real roots. This gives a purely linear-algebraic way to count real roots.
Goal
Implement the Hermite matrix of a univariate polynomial, compute its signature exactly, and compare the resulting real root counts, in correctness and speed, with Sturm sequences.