The Gauss Lucas Theorem

The Gauss Lucas Theorem
Matt Boelkins

One of the most famous results in first semester calculus is Rolle's Theorem, which is a special case of the Mean Value Theorem:

Rolle's Theorem. If f is a continuous function on [a,b], differentiable on (a,b), and f(a) = f(b) = 0, then there exists some point c in (a,b) such that f'(c) = 0.

Such a point c, of course, is called a critical number.

In what follows, we will be interested in a special (familiar) class of differentiable functions: polynomials. For real polynomial functions,

p(z) = zⁿ + a_n_-1zⁿ^-1+ ... + a₁z + a₀,

where all the constants a_i are real, from Rolle's Theorem it follows that between any two real zeros of p, there exists a real critical number (that is, a real zero of p'). An interesting topic of ongoing research is centered on the question: how are the critical numbers of a polynomial with all real zeros distributed relative to the zeros of the function.

The situation is more complicated when some of the zeros of p are complex. For example, consider f(z) = (z² - 1)(z - i sqrt(3)); its zeros are the vertices of an isosceles triangle (-1,0), (1,0), and (0,i sqrt(3). It follows that f'(z) = 3(z - i/sqrt(3))² so that f has a repeated critical point at i/sqrt(3), a point interior to the triangle. This shows that the critical points (even of a polynomial function) are not guaranteed to lie "between" a pair of roots, and thus Rolle's Theorem does not admit a direct complex extension. While there is no direct analog of Rolle's Theorem, there are results that discuss the location of critical numbers relative to the polynomial's roots. Perhaps the most famous of these theorems in the Gauss-Lucas Theorem, discovered in the 19th century by F. Lucas:

Gauss-Lucas Theorem. The convex hull of the zeros of a polynomial p contains all the zeros of its derivative, p'.

Briefly, a set is convex if and only if a line segment joining any two points in the set is itself a subset of the set. Moreover, given any set S, the convex hull of the set, H(S), is the smallest convex set containing S. Thus, what the Gauss-Lucas Theorem is saying is that somehow all the zeros of p' lie "inside" the zeros of p.

The picture below shows this: we have an image of the convex hull of the zeros of a polynomial p containing the convex hull of the zeros of p', which in turn contains the convex hull of the zeros of p''.

Observe carefully that the Gauss-Lucas Theorem makes a claim about a relationship between two sets of points, while Rolle's Theorem really only makes a claim about a single critical number. We will thus distinguish between Rolle-type statements (which involve just two of the zeros of a polynomial), and Hausdorff-type statements (which involve the set of all zeros of the polynomial and the set of all its critical points). To find out why we call these statements ``Hausdorff-type'', you might want to read about Professor Schlicker's research area.