University:
Massachusetts Institute of Technology
Course:
18.900 | Geometry and Topology in the Plane
Academic year:

2023
Views:

435

Pages:

4
Author:

Zayden Mercer

14. Complex polynomials Given how long we’ve been talking about the plane, it’s surprising that complex numbers haven’t appeared so far. We now fix this: • One can use winding numbers to detect zeros of complex polynomials. • Unlike the case of real equations, this is an “if and only if” process, and provides a count of how many zeros lie in a disc of radius r, assuming that the zeros are counted with their proper positive multiplicities. (14a) Complex numbers. A complex number is given by its real and imaginary parts, z = x + iy, hence is the same as a point (x, y) in the plane. One writes |z| instead of ∥z∥ for its length, meaning p (14.1) |z| = x2 + y 2 for z = x + iy. There’s a famous formula for trigonometric functions in terms of the complex exponential, (14.2) eiθ = cos(θ) + i sin(θ). This means that complex numbers are written in radial coordinates as z = reiθ . As one sees from that, the product of complex numbers multiplies the radii and adds the angles: (14.3) (r1 eiθ1 )(r2 eiθ2 ) = (r1 r2 )ei(θ1 +θ2 ) . One can think of smooth loops as taking values in complex numbers, meaning c(t) ∈ C. The simplest example may be the loop (14.4) c(t) = eint , with T = 2π, for some integer n. This goes n times around the radius 1 circle (if n is negative, that means clockwise). One can see this directly, eint = cos(nt) + i sin(nt); or one can say that eit goes once around the circle, and then taking the n-th power has the effect of multiplying the angles by n. (14b) Roots and multiplicities. Take a complex polynomial of degree n: (14.5) f (z) = an z n + an−1 z n−1 + · · · + a1 z + a0 , withan ̸= 0. The fundamental theorem of algebra says that we can always write this as (14.6) f (z) = a(z − w1 )m1 · · · (z − wk )mk , where a = an is the leading coefficient; the roots wi are all different from each other; and the multiplicities mi are positive integers, whose sum is n. This is a theoretical existence statement, which basically says that f has n zeros once those are counted with the proper multiplicities. If we know w is a root, we can actually compute its multiplicity without writing the polynomial in the form described above: Lemma 14.1. The multiplicity of a root w is the smallest m such that the m-th derivative of f at w is nonzero. Example 14.2. Take f (z) = z + 3z 2 − 3z 3 + z 4 , which satisfies f (1) = 0. We compute (14.7) f ′ (z) = −1 + 6z − 9z 2 + 4z 3 , f ′ (1) = 0, f ′′ (z) = 6 − 18z + 12z 2 , f ′′ (1) = 0, f ′′′ (z) = −18 + 24z, f ′′′ (1) = 6 ̸= 0, so the multiplicity at 1 is 3. From now on, we write mult(f, w) for the multiplicity of f at w (if w is not a root, one can set that multiplicity to 0). (14c) The winding number formula. Applying the same idea as in the previous lecture, we look at the image of a circle of radius r > 0 under f , which is the loop (with T = 2π) (14.8) d(t) = f (reit ) = f (r cos(t) + ri sin(t)). Suppose that f has is no root on the circle of radius r around the origin, so that the winding number wind(d, 0) is defined. Then: Theorem 14.3. For a loop (14.8), (14.9) X wind(d, 0) = mult(f, w), |w| 0 if and only if there is a solution of f (w) = 0 inside the circle. This two-way implication is part of the special magic of the class of holomorphic functions, of which polynomials are the simplest examples. Example 14.4. Take f (z) = z 5 − z 3 − 12 . The loop d(t) = f (eit ) = e5it − e3it − 1 2 looks like this: 2 1 (14.10) -2 1 -1 -1 -2 From that, one reads off the winding number around the origin, wind(d, 0) = 3 (the picture doesn’t tell you which way the loop goes; but the other direction gives a winding number of −3, which is impossible). This means that we have three possibilities: either there are three solutions of f (p) = 0 with |p| < 1, each having multiplicity 1; or two solutions, with multiplicities 1, 2; or a single solution, with multiplicity 3 (in fact, the first is the case, but you can’t tell that just from our computation). Example 14.5. Take f (z) = (z + i)(z − i)(z + 1)(z − 1)(z − 1/4). There is one root with |p| = 1/4, and four roots with |p| = 1. All roots have multiplicity 1. Consequently, the winding number wind(d, 0) remains zero for r < 1/4, and then jumps to 1. The jump happens in a relatively simple way, by d moving across the origin: 0.2 0.2 0.2 0.1 0.1 0.1 (14.11) 0.2 0.3 0.4 0.1 0.1 0.5 0.2 0.3 0.4 0.1 0.5 0.2 0.3 0.4 0.5 -0.1 -0.1 -0.1 -0.2 -0.2 -0.2 r = 0.23 r = 0.25 r = 0.27 The winding number remains at that value for 1/4 < r < 1, and then jumps to 5 when crossing r = 1. At that value, four parts of the loop d all pass through the origin simultaneously: 2 2 2 1 -1.0 1 0.5 -0.5 1.0 1.5 2.0 1 1 -1 2 1 -1 2 (14.12) -1 -1 -1 -2 -2 -2 r = 0.98 r=1 r = 1.03 Example 14.6. Take f (z) = (z − 12 )3 (z − i). This has a root of multiplicity 3 at 1/2, and a root of multiplicity 1 at i. Correspondingly, we expect the winding number to be 0 for r < 1/2, then 3 for r ∈ (1/2, 1), and finally 4 for r > 1. The jump from 0 to 3 comes with a sudden curling behaviour: 0.35 3 6 2 4 0.30 0.25 0.20 (14.13) 2 1 0.15 -6 -3 0.10 -2 1 -1 -4 2 -2 4 2 -2 0.05 -1 -4 -0.2 -0.1 r = 0.2 0.1 0.2 r = 0.8 r = 1.1 (14d) Other values. We have focused on the equation p(z) = 0, but simply by subtracting a constant from p, one can apply the result to equations p(z) = u for an arbitrary complex number u. Corollary 14.7. Take d(t) = p(reit ) as before. For every u where it is defined, the winding number wind(p, u) is nonnegative; and it is > 0 if and only if there is a solution of p(z) = u with z inside the circle of radius r. Example 14.8. Take the leftmost picture from Example 14.5, with r = 0.98. We have not drawn that, but the motion of the loop is anticlockwise (meaning, to the left at its topmost point). As a consequence, one can check that the winding numbers are positive for all regions except the outermost infinite region. It follows that for any u lying in those regions, the equation p(z) = u has a solution with |z| < 0.98. Corollary 14.9. Take d1 = p(r1 eit ), d2 = p(r2 eit ), for some r2 > r1 > 0. Then, for every complex number u where both winding numbers are defined, we have wind(d2 , u) ≥ wind(d1 , u). (14.14) In other words, as the radius increases, the winding number of our loops around any point can only go up or stay the same; it can never go down. (14e) Proof. The proof of the theorem is, as usual for loops, by a deformation argument. We write our polynomial as a product, but separating the roots that lie inside and outside the circle of radius r: Y Y (14.15) f (z) = (z − wi )mi )(z − wi )mi . |wi |r Now we introduce a parameter s ∈ [0, 1] which changes it like this: Y Y (14.16) fs (z) = (z − swi )mi (sz − wi ) . |wi |r If |wi | < r is a root of f , then s|wi | is a root of fs . In the second instance, if |wi | > r is a root of f , then |wi |/s is a root of fs (or for s = 0, there is no corresponding root). In words, the roots lying inside the circle of radius r move inwards as s becomes smaller, and those lying outside the circle move to infinity. At no time s does fs actually have a root on the circle. Therefore, if we define ds (t) = fs (reit ), (14.17) then wind(ds , 0) remains the same for all s. For s = 1 we have f1 = f , so d1 = d is the loop associated to the original polynomial. Now let’s see what we get for s = 0: Y Y (14.18) f0 (z) = z mi (−wi )mi = z m a, |wi |r mi where m = |wi |r (−wi ) is a nonzero constant. The associated loop is d0 (t) = eimt a, which goes m times around the circle of radius |c|. Therefore, wind(d0 , 0) = m. The same is therefore true of wind(d, 0) = wind(d1 , 0). Looking at the definition of m, that’s exactly what the theorem says!

Complex Polynomials

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