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November 29, 2013

Critical points of polynomials with respect to the roots

Filed under: mathematics — ckrao @ 1:28 pm

Recall that a critical point of a polynomial p:\mathbb{C} \rightarrow \mathbb{C} is a point w for which p'(w) = 0. The Gauss-Lucas theorem states that the critical points of p all lie within the convex hull of its set of zeros and we shall explain why this holds below. The real-number analogue of this is that the stationary points of a polynomial with real roots all lie between the smallest and largest root.

Denote the roots of the polynomial by z_1, z_2, \ldots, z_n (we’ll assume they are distinct) and the critical points by w_1, w_2, \ldots, w_{n-1}. We then may write our polynomial as p(z) = k \prod_{i=1}^n (z - z_i) for some constant k and its derivative by the product rule is

\displaystyle p'(z) = k \sum_{j=1}^n \prod_{i=1}^n \frac{z-z_i}{z-z_j} = p(z) \sum_{j=1}^n \frac{1}{z-z_j}.\quad \quad ...(1)

If p(w_i) \neq 0 then as w_i is a critical point of p, p'(w_i) = 0 and so

\displaystyle 0 = \sum_{j=1}^n \frac{1}{w_i-z_j} = \sum_{j=1}^n \frac{\bar{w_i} - \bar{z_j}}{|w_i - z_j|^2},

from which

\displaystyle w_i \sum_{j=1}^n \frac{1}{|w_i - z_j|^2} = \sum_{j=1}^n \frac{z_j}{|w_i - z_j|^2}.\quad\quad...(2)

In other words,

\displaystyle w_i = \sum_{k=1}^n \alpha_k z_k,\quad\quad...(3)

where \alpha_k = \frac{1/|w_i - z_k|^2}{\sum_{j=1}^n 1/|w_i - z_j|^2}.

Since \sum_{k=1}^n \alpha_k = \frac{\sum_{k=1}^n 1/|w_i - z_k|^2}{\sum_{j=1}^n 1/|w_i - z_j|^2} = 1, w_i is a convex combination of the roots z_k of p(z). In other words, w_i is in the convex hull of the set of roots z_k.

In the other case of p(w_i) = 0, w_i is equal to one of the roots and so is also in the convex hull of the set of roots. This completes the proof of the theorem.

However as mentioned in [1] an interpretation of (3) by Cesàro in 1885 is that if we fix unit masses at the roots z_i (if there are repeated roots place mass equal to the multiplicity), the critical points will be at precisely those points experiencing zero net force, assuming particles repel with a force proportional to the inverse of the distance between them (as opposed to an inverse square law). To see why this is so, if the force at a point w due to a mass at z is proportional to the vector \frac{1}{|z-w|^2}(z-w) (an inverse law, as this has magnitude 1/|z-w|), then the net force is the sum of these over the masses at all the roots. In terms of any critical point w_i, this is the following:

\begin{aligned} \sum_{k=1}^n \frac{1}{|z_k-w|^2}(z_k-w) &= \sum_{k=1}^n \frac{1}{|z_k-w|^2}z_k- w\sum_{k=1}^n \frac{1}{|z_k-w|^2}\\&= w_i\sum_{k=1}^n \frac{1}{|z_k-w|^2}- w\sum_{k=1}^n \frac{1}{|z_k-w|^2} \quad \text{by (2)}\\&=(w_i-w)\sum_{k=1}^n \frac{1}{|z_k-w|^2}, \end{aligned}

which is zero if and only if w=w_i, i.e. the point having zero net force is a critical point of p.

Bôcher’s theorem generalises this result, replacing polynomials with rational functions (ratios of polynomials) and simply applying negative unit masses at poles.

Another interesting result, mentioned in [2], is that the product of the distances from a root z_1 to the other roots (assuming all distinct roots) is precisely n times the product of the distances from z_1 to the critical points w_j. That is,

\prod_{i=2}^n |z_1 - z_i| = n \prod_{j=1}^{n-1} |z_1 - w_j|.\quad\quad ...(4)

To see this, if p(z) = k \prod_{i=1}^n (z-z_i), p'(z) will have the form kn \prod_{j=1}^{n-1} (z-w_j) (since it has leading term nz^{n-1}. Note that z_1 is not a critical point since we assume the roots are distinct. Hence we may consider the quotient

\begin{aligned} \frac{\prod_{i=2}^n (z_1 - z_i)}{\prod_{j=1}^{n-1} (z_1 - w_j)} &= \frac{\lim_{z \rightarrow z_1} p(z)/(z-z_1)}{p'(z_1)/n}\\&= \lim_{z \rightarrow z_1} \frac{np(z)}{(z-z_1)p'(z)}\\ &= \lim_{z \rightarrow z_1} \frac{np(z)}{(z-z_1)p(z) \sum_{j=1}^n 1/(z-z_j)} \quad \text{from (1)} \\&= \lim_{z \rightarrow z_1} \frac{n}{\sum_{j=1}^n (z-z_1)/(z-z_j)} \\ &= \lim_{z \rightarrow z_1} \frac{n}{1 + \sum_{j=2}^n (z-z_1)/(z-z_j)}\\ &= n.\end{aligned}

Taking magnitudes of both sides leads to the desired result (4). One also can obtain an angular relationship by taking the argument of both sides.

The critical points for the case n = 3 have a beautiful geometric interpretation as described in [3]: from the triangle formed by the roots, the critical points are the foci of the largest ellipse that is inscribed in the triangle (this ellipse also happens to pass through the triangle’s midpoints) – this is now known as Marden’s theorem. A generalisation proved by Marden in 1945 and mentioned in [1] is that the critical points of an n-degree polynomial are the foci of a degree n-1 plane curve tangent to each of the n(n-1)/2 line segments formed by taking pairs of the n roots. The points of tangency divide each line segment into a ratio corresponding to the multiplicities of the roots at the endpoints. Also, if the n-dimensional polygon is a linear (affine) transform of a regular polygon, it will have an inscribed ellipse passing through its midpoints (called the Steiner inellipse) and the foci of that ellipse will correspond to critical points.

Below are some other nice results about the critical points of a polynomial, gathered from [4] (also see references therein):

  • The centre of mass of the roots of p coincides with the centre of mass of the critical points of p (a nice exercise to prove).
  • (Anderson) The polynomial root dragging theorem for polynomials with real distinct roots: as roots are dragged to the right by at most \epsilon (so as not to coincide), the critical points also move to the right, each by less than \epsilon.
  • (Boelkins et al., later Frayer) The polynomial root squeezing theorem for polynomials with real distinct roots: if two roots z_i, z_j are squeezed together by the same amount without passing other roots, the critical points move towards (z_i + z_j)/2 or stay fixed.
  • (Peyser) Critical points are not too close to the roots: if a polynomial has only real roots z_1 < z_2 < \ldots < z_n and critical points w_1 < w_2 < \ldots w_{n-1} the critical points satisfy
    \displaystyle z_k + \frac{z_{k+1} - z_k}{n - k + 1} \leq w_k \leq z_{k+1} - \frac{z_{k+1} - z_k}{k+1}, \quad k= 1, \ldots, n-1.
    Equality cases are satisfied by Bernstein polynomials. Replacing 1/(n-k+1) and 1/(k+1) by m_k/(n-k+m_k) and m_{k+1}/(k + m_{k+1} respectively gives a corresponding result (due to Melman) taking into account multiplicities m_k for root z_k.

References

[1] Qazi Ibadur Rahman, Gerhard Schmeisser, Analytic Theory of Polynomials, Oxford University Press, 2002.

[2] Maxime Bocher, Some Propositions Concerning the Geometric Representation of Imaginaries, Annals of Mathematics, Vol. 7, No. 1/5 (1892 – 1893), pp. 70-72.

[3] Dan Kalman, The Most Marvelous Theorem in Mathematics,  The Journal of Online Mathematics and Its Applications, Vol. 8, March 2008, Article ID 1663.

[4] Neil Biegalle, Investigations in the Geometry of Polynomials, McNair Scholars Journal, Volume 13, Issue 1, Article 3. Available at: http://scholarworks.gvsu.edu/mcnair/vol13/iss1/3

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