Gauss-Lucas Theorem
Gauss-Lucas Theorem: If P(z) is a polynomial with complex coefficients, then all zeros of its derivative P'(z) lie within the convex hull of the zeros of P(z).
Specht's Theorem (1959): The zeros of P'(z) actually lie in the convex hull of the Specht points pjk = (cj + (n-1)ck)/n, which is a strictly smaller region!
Zeros of P'(z) (number = multiplicity)
$$P(z) = \text{Click to add zeros...}$$
$$P'(z) = —$$
Interactive Demo: Click to add zeros or drag existing ones to move them. The visualization shows:
- Blue dashed region: Gauss-Lucas hull - convex hull of the zeros
- Purple points: Specht points pjk = (cj + (n-1)ck)/n
- Purple solid region: Specht hull - convex hull of Specht points (strictly smaller!)
- Red dots: Zeros of P'(z), always inside the purple Specht hull
- Concentric rings + number: Multiple roots (e.g., "3" means triple root)
Notice how Specht's theorem gives a tighter bound than Gauss-Lucas! For the square example, the derivative has a triple root at the center.