Zeros of a Family of Complex-Valued Harmonic Trinomials

Abstract

It is known that complex-valued harmonic polynomials of degree n can have more than n zeros. In the 2020 paper “Zeros of a One-Parameter Family of Harmonic Trinomials” by Brilleslyper et al., the authors consider a one-parameter family of complex-valued harmonic polynomials and determine, for different values of the real parameter, the number of zeros. We consider the same family but allow the parameter to be complex. As in the previous paper, our proof relies on the Argument Principle for Harmonic Functions and again requires us to find the winding number about the origin of a hypocycloid. The geometry is more complicated in this case, however. This additional complexity is reflected in the main theorem, which shows that the number of transitions in the number of zeros and the nature of these transitions depends on the argument of the complex parameter.

1 Introduction

We consider the number of zeros of a family of complex-valued harmonic polynomials, that is, polynomials of the form $f=u+iv$ where u, v are real-valued harmonic polynomials. Such polynomials can also be written as the sum of an analytic polynomial and the conjugate of an analytic polynomial. Thus, we may write $f=h+{\overline{g}}$ for h, g analytic. For analytic polynomials, the Fundamental Theorem of Algebra readily gives the number of zeros. For complex-valued harmonic polynomials, the situation is more complicated because the number of zeros can be larger than the degree of the polynomial and can vary as the coefficients vary. In this paper, we consider the family of complex-valued harmonic trinomials

$$\begin{aligned} p_c(z)=z^n+c{\overline{z}}^k-1, \end{aligned}$$

(1.1)

where $1 \le k \le n-1$, $n \ge 3$, $c \in \mathbb {C}$, and $\gcd (n,k)=1$. We are interested in how the number of zeros of $p_c$ changes as c varies. Our theorem generalizes the work of Brilleslyper, Brooks, Dorff, Howell, and Schaubroeck [1] in which they consider the family (1.1) for positive real c. As in the theorem in [1], our theorem states that the number of zeros increases from n to $n+2k$ as the value of $|c| = \rho$ increases. However, we show that the particular values of $\rho$ at which the number of zeros increases and the number of such values depend upon the argument of c. These $\rho$ values at which the number of zeros increases are called the critical $\rho$ values.

More specifically, our main theorem is as follows:

Theorem 1.1

Let $p_c(z)=z^n+c{\overline{z}}^k-1$, where $1 \le k \le n-1$, $n \ge 3$, $\gcd (n,k)=1$, and $c=\rho e^{i \tau }$. Let

$$\begin{aligned} N_{\tau } = {\left\{ \begin{array}{ll} \lfloor k/2 \rfloor + 1 &{} \text {if}\ \tau = \frac{m \pi }{n}\ \text {with}\ m\ \text {even,} \\ \lfloor (k+1)/2 \rfloor &{} \text {if}\ \tau = \frac{m \pi }{n}\ \text {with}\ m\ \text {odd,} \\ k &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

(1.2)

For each $\tau$, there are $N_{\tau }$ critical values $\rho _j (\tau )$, with $0<\rho _1 (\tau )<\rho _2 (\tau )< \cdots < \rho _{N_\tau } (\tau )$, such that

• (a) if $0 \le \rho <\rho _1 (\tau )$, then $p_c$ has n distinct zeros,

• (b) if $\rho _j (\tau )< \rho < \rho _{j+1} (\tau )$ for some $1 \le j \le N_\tau -1$, then $p_c$ has

  • (i) $n+4j-2$ distinct zeros if $\tau = \frac{m \pi }{n}$ with m even,

  • (ii) $n+4j$ distinct zeros if $\tau = \frac{m \pi }{n}$ with m odd, and

  • (iii) $n+2j$ distinct zeros otherwise, and

• (c) if $\rho >\rho _{N_\tau } (\tau )$, then $p_c$ has $n+2k$ distinct zeros.

In the preprint [5], the authors show that the zeros of $p_c$ lie in the annulus given by $(|c| - 1)^\frac{1}{n-k} \le |z| \le ( |c|+1)^\frac{1}{n-k}$ if z is a zero of $p_c$ with $|z| \ge 1$ and $c \in {\mathbb {C}}$. In this paper, we do not discuss the locations of the zeros.

We prove our main theorem in Sect. 3. We illustrate the theorem with an example.

Example 1.2

Consider $p_c(z) = z^5 + c{\bar{z}}^4 - 1$ for $c=\rho e^{i\frac{\pi }{3}}$. Because $\tau =\frac{\pi }{3}$, in the notation of Theorem 1.1, $N_\tau = 4$. Thus, the number of zeros strictly increases 4 times so that there are 5 possible numbers of zeros that are all distinct. Part (a) tells us that the polynomial has 5 distinct zeros when $\rho$ is between 0 and $\rho _1$. By part (b), there are 7, 9, and 11 distinct zeros for $\rho _1< \rho < \rho _2$, $\rho _2< \rho < \rho _3$, and $\rho _3< \rho < \rho _4$, respectively, and by part (c), there are 13 distinct zeros for $\rho _4 < \rho$. See Fig. 1.

Fig. 1

Zeros of $p_c(z) = z^5 + c{\overline{z}}^4 - 1$ and the critical circle $\Gamma _c$ for various values of c

The circle in each figure is the critical circle; it is essential to our zero-counting arguments and will be discussed in the next section.

2 Background

As in [1], we use the Argument Principle for complex-valued harmonic functions. In order to state this principle, we need to discuss in more detail the geometry of complex-valued harmonic functions. Recall that an analytic function is conformal and thus sense-preserving at each point where the derivative does not vanish. However, complex-valued harmonic polynomials may not be sense-preserving. The complex dilatation $\omega (z)$ measures how far the function f is from being sense-preserving. If $f(z) = h(z) + \overline{g(z)}$, then $\omega (z) = \frac{g'(z)}{h'(z)}$. One can show that f is sense-preserving when $|\omega (z)|<1$ and sense-reversing when $|\omega (z)|>1$. The critical curve is the set of points where $|\omega (z)| = 1$.

Recall that the Argument Principle for analytic functions enables us to count zeros (with multiplicity) inside a curve. We need an analog to the multiplicity or order of a zero for complex-valued harmonic functions. Recall that an analytic function f has a zero of order m at $z_0$ if in the Taylor series expansion $f(z) = \sum _{j=0}^\infty a_j (z-z_0)^j$ about $z_0$, $a_j=0$ for $j<m$ and $a_m\ne 0$. Similarly, if the complex-valued harmonic function f has a zero at $z_0$, we write

$$\begin{aligned} f(z) = h(z) + \overline{g(z)} = a_0+\sum _{j=r}^\infty a_j (z-z_0)^j + \overline{\left( b_0+\sum _{j=s}^\infty b_j (z-z_0)^j\right) }, \end{aligned}$$

(2.1)

where $a_r, b_s \ne 0$ and $r, s>0$; then $z_0$ has order r if it is in the sense-preserving region and order $-s$ if it is in the sense-reversing region. If $z_0$ is on the critical curve, its order is undefined. The main difference in the complex-valued harmonic case is that a negative order indicates that the zero is in the sense-reversing region. For further information about complex-valued harmonic functions, see [2,3,4].

We are now prepared to understand and apply the Argument Principle for complex-valued harmonic functions:

Theorem 2.1

(Argument Principle for Complex-Valued Harmonic Functions) Let D be a Jordan domain with boundary C. Suppose that f is harmonic on D, continuous on ${\overline{D}}$, $f \ne 0$ on C, and there are no zeros $z_0$ of f in D for which $|\omega (z_0)|=1$. Then, the total change in the argument of f as C is traversed in the positive direction is $2 \pi N$, where N is the sum of the orders of the zeros of f in D.

For the family $p_c$ defined in (1.1), the critical curve is the set of all points satisfying $|\omega (z)| = 1$, which by direct calculation occurs when $|z| = \left( \frac{\left|c \right| k}{n}\right) ^{\frac{1}{n-k}}$. Note that $p_c$ is sense-reversing inside the circle and sense-preserving outside. The critical circle plays a central role in our analysis; thus, we define

$$\begin{aligned} \Gamma _c=\left\{ z:\left|z \right|=R_c = \left( \frac{\left|c \right|k}{n}\right) ^{\frac{1}{n-k}}\right\} . \end{aligned}$$

(2.2)

Consider again Fig. 1, which shows the critical circle $\Gamma _c$ for several different values of c. Note in particular that whenever a new zero appears inside $\Gamma _c$ (i.e., in the sense-reversing region), a new zero also appears outside $\Gamma _c$ (in the sense-preserving region). Thus although the total number of zeros is not preserved as $\rho$ increases, the difference in the number of zeros in the two regions is preserved. This fact hints at the correct generalization of the Fundamental Theorem of Algebra, given in Lemma 3.1 below.

3 Proof of Theorem 1.1

We prove Theorem 1.1 using a series of lemmas. Lemma 3.1 generalizes the Fundamental Theorem of Algebra by showing that the sum of the orders of the zeros of $p_c$ is n. Lemma 3.2 simplifies our task by concluding that the order of each zero is either 1 or $-1$. These lemmas prepare us to examine the number of zeros in the sense-reversing region by considering $p_c(\Gamma _c)$ and its winding number about the origin. Lemma 3.4 classifies this image as a hypocycloid centered at $-1$, and Lemmas 3.1 and 3.2 show that the number of zeros changes only when the increase in zeros in the sense-reversing region is balanced by an equal increase in the sense-preserving region. Lemmas 3.6 and 3.7 investigate the geometry of the hypocycloid and are essential for calculating its winding number $W_c$ about the origin. Lemma 3.6 helps us compute $W_c$ for sufficiently large $\rho$ and Lemma 3.7 helps us determine how $W_c$ depends on c for intermediate values of $\rho$. Finally, Lemma 3.8 finds the values of $N_\tau$ and $W_c$ for different values of $\tau$. These lemmas are combined to prove Theorem 1.1.

Lemma 3.1

Let $p_c$ be as in (1.1). For R sufficiently large, the winding number of the image of $|z|=R$ under $p_c$ around the origin is n. Thus, for all c, the sum of the orders of the zeros of $p_c$ not on $\Gamma _c$ is n.

Proof

The proof follows as in [1] from a standard Rouché-type argument by comparing $|p_c(Re^{it})-R^ne^{int}|$ to $|R^ne^{int}|$ for sufficiently large R. $\square$

Next, we show that the order of each zero is either 1 or $-1$ using the same approach as in [1]. This fact implies that counting the total number of zeros is equivalent to counting the total number of distinct zeros.

Lemma 3.2

All zeros of $p_c(z) = z^n + c{\overline{z}}^k - 1$ not on $\Gamma _c$ have order 1 or $-1$.

Proof

Observe that $p_c(z)=h(z)+\overline{g(z)}=(z^n-1)+\overline{({\bar{c}}z^k)}$. Let $z_0$ be a zero of $p_c$. We know that $z_0\ne 0$ since $p_c(0)=-1$. Using the notation and results from (2.1) and the associated discussion, we need only show that $r=s=1$. We calculate that $a_1 = h'(z_0) = nz_0^{n-1} \ne 0$ and $b_1 = g'(z_0) = {\bar{c}}kz_0^{k-1} \ne 0$. So $r=s=1$ and $z_0$ has order $+1$ for $|z_0|>R_c$ and order $-1$ for $|z_0|<R_c$. $\square$

The main idea is to count zeros in the sense-reversing region by finding the winding number of the image of the critical curve under $p_c$. We show that $p_c(\Gamma _c)$ is a rotation of a hypocycloid of type $(n+k, n)$ as in the following definition.

Definition 3.3

([7]) A hypocycloid centered at the origin is the curve traced by a fixed point on a circle of radius b rolling inside a larger origin-centered circle of radius a. The curve is given by the parametric equations

$$\begin{aligned} x(\phi )&= (a-b)\cos (\phi ) + b \cos \left( \frac{a-b}{b}\phi \right) , \end{aligned}$$

(3.1)

$$\begin{aligned} y(\phi )&= (a-b)\sin (\phi ) - b \sin \left( \frac{a-b}{b}\phi \right) . \end{aligned}$$

(3.2)

If the ratio $\frac{a}{b}$ is written in reduced form as $\frac{p}{q}\in {\mathbb {Q}}$, then the hypocycloid has p cusps, and each arc connects cusps that are q away from each other in a counterclockwise direction. Such a hypocycloid is called a (p, q) hypocycloid, and the range of $\phi$ values to trace the entire hypocycloid is $0\le \phi \le 2\pi q$.

A hypocycloid with a cusp lying on the positive real axis as in Definition 3.3 will be called a position zero hypocycloid.

Combining the parametric equations into $H(\phi ) = x(\phi ) + iy(\phi )$, we see that

$$\begin{aligned} H(\phi ) = (a-b) e^{i\phi } + b e^{i\phi \left( \frac{b-a}{b}\right) } \end{aligned}$$

(3.3)

is an equivalent way to describe a hypocycloid in the complex plane. We use this form for the following lemma, which is a generalization of [1, Lemma 2.4].

Lemma 3.4

As above, let $c = \rho e^{i \tau }$. The image $p_c(\Gamma _c)$ of the critical curve is an $(n+k, n)$ hypocycloid centered at $z = -1$ that has been rotated by an angle of $\frac{\tau n}{n+k}$.

Proof

Recall that

$$\begin{aligned} R_c = \left( \frac{\rho k}{n}\right) ^{\frac{1}{n-k}}. \end{aligned}$$

(3.4)

Then

$$\begin{aligned} p_c(\Gamma _c) = p_c\left( R_c e^{i\theta }\right) = R_c^n e^{in\theta } + \rho R_c^k e^{i(\tau - k\theta )} - 1. \end{aligned}$$

(3.5)

We translate to the right 1 and show the resulting curve is a position zero hypocycloid multiplied by $e^{i \alpha }$ for some angle $\alpha$. We let

$$\begin{aligned} a = \rho ^{\frac{n}{n-k}} \left( \frac{k}{n}\right) ^{\frac{k}{n-k}} \left( \frac{n+k}{n}\right) \qquad \text {and}\qquad b = \rho ^{\frac{n}{n-k}} \left( \frac{k}{n}\right) ^{\frac{k}{n-k}} = \rho R_c^k. \end{aligned}$$

Thus

$$\begin{aligned} a - b = \rho ^{\frac{n}{n-k}} \left( \frac{k}{n}\right) ^{\frac{n}{n-k}} = R_c^n \qquad \text {and}\qquad \frac{a-b}{b} = \frac{k}{n}. \end{aligned}$$

(3.6)

We immediately see that $p_c(\Gamma _c) + 1 = (a-b) e^{in\theta } + b e^{i(\tau - k\theta )}$. Set $\phi = n\theta - \frac{\tau n}{n+k}$ so that $\theta = \frac{1}{n} \phi + \frac{\tau }{n+k}$. Then,

$$\begin{aligned} p_c(\Gamma _c) + 1 = (a-b) e^{i\left( \phi + \frac{\tau n}{n+k}\right) } + b e^{i\left( \tau - \frac{k}{n} \phi - \frac{\tau k}{n+k}\right) }. \end{aligned}$$

(3.7)

Because $\frac{k}{n} = \frac{a-b}{b}$, $\frac{k}{n+k} = \frac{a-b}{a}$, and $\frac{n}{n+k} = \frac{b}{a}$, we see that

$$\begin{aligned} p_c(\Gamma _c) + 1&= (a-b) e^{i\left( \phi + \frac{\tau b}{a}\right) } + b e^{i\left( \tau - \frac{a-b}{b} \phi - \frac{\tau (a-b)}{a}\right) } \\&= (a-b) e^{i\left( \phi + \frac{\tau b}{a}\right) } + b e^{i\left( \frac{b-a}{b} \phi + \frac{\tau b}{a}\right) } \\&= e^{i\frac{\tau b}{a}} \left( (a-b) e^{i \phi } + b e^{i \phi \left( \frac{b-a}{b}\right) } \right) . \end{aligned}$$

This is the equation of a hypocycloid rotated by the angle $\alpha = \frac{\tau b}{a} = \frac{\tau n}{n+k}$. Note that when $\phi = 0$, $p_c(\Gamma _c) +1 = a e^{i \frac{\tau b}{a}}$, so we will always begin graphing at the angle $\frac{\tau n}{n+k}$, where the first cusp of the hypocycloid sits. Furthermore, the entire hypocycloid is traced for $0 \le \phi \le 2\pi n$. Adding back in the $-1$, we see that $p(\Gamma _c)$ is a hypocycloid centered at $-1$ which has been rotated around this point. $\square$

The angle of rotation $\alpha$ of the hypocycloid directly affects how the winding number $W_c$ changes as $|c| = \rho$ increases. Consequently, the number of zeros in the sense-reversing region changes differently for different values of $\alpha$. We illustrate the three cases of $\alpha$ and the $\tau$-values that produce them. These are the three cases in part (b) of Theorem 1.1. The three hypocycloids in Fig. 2 correspond to these three cases for $p_c(z) = z^5 + c {\overline{z}}^4 -1$.

Fig. 2

$p_c(\Gamma _c)$ for the polynomial $p_c(z) = z^5 + c{\overline{z}}^4 - 1$ for various angles of rotation $\alpha$ induced by corresponding $\tau$

We discuss these cases further in the next few lemmas.

As in [1], we have $\frac{b}{a}>\frac{1}{2}$, which makes the smaller of the two circles forming the hypocycloid have radius larger than half that of the larger circle, meaning that the hypocycloid traces clockwise around its center while the inner circle traverses counterclockwise.

We can extend the observations about hypocycloid geometry made in [1] to a more general case of rotated hypocycloids. As before, cusps which are q apart in a counterclockwise direction may also be considered $p-q$ apart in the clockwise direction.

As in [1], we may use the winding number $W_c$ of the hypocycloid $p_c(\Gamma _c)$ to determine the number of zeros in the sense-reversing region. Because the furthest point from the center of a hypocycloid is at a distance from the center depending on $\rho$ (with an increase in $\rho$ yielding an increase in this distance), for each $\tau$, there exists $\rho _1(\tau )$, such that all intersections of the hypocycloid with the real axis are to the left of the origin if and only if $\rho < \rho _1(\tau )$. Then, $W_c = 0$ whenever $\rho < \rho _1(\tau )$. As $\rho$ increases, $W_c$ is unchanging until $\rho = \rho _1(\tau )$, at which point the winding number decreases, becoming negative. The exact winding number depends on $\tau$. As $\rho$ increases further, it passes through other critical values $\rho _1(\tau )< \rho _2(\tau )< \ldots < \rho _{N_\tau }(\tau )$, each dependent on $\tau$. These are those $\rho$-values for which the hypocycloid crosses the origin; these are associated with the intersections of the hypocycloid with the real axis right of its center, since such intersections move to the right and eventually pass through the origin as $\rho$ increases. Consequently, we must determine the number of intersections $N_\tau$ of the hypocycloid with the real axis to the right of its center for each $\tau$, as well as the uniqueness of such intersections, in order to understand how $W_c$ and the number of zeros of $p_c$ change.

Example 3.5

We apply Lemma 3.4 and the previous discussion to $p_c(z) = z^5 + c{\overline{z}}^4 - 1$. Figure 3 shows the (9, 5) hypocycloids $p_c(\Gamma _c)$, fixing $\tau = \frac{\pi }{3}$ and increasing $\rho$. Lemma 3.4 indicates that each hypocycloid is centered at $-1$ and is rotated by $\frac{\tau n}{n+k} = \frac{5\pi }{27}$.

Fig. 3

$p_c(\Gamma _c)$ for the polynomial $p_c(z) = z^5 + c{\overline{z}}^4 - 1$ and three values of c

As $\rho$ increases, $p_c(\Gamma _c)$ produces larger hypocycloids. We will refer to Example 1.2 to show that the hypocycloid’s growth is tied to the change in number of zeros. For small $\rho$, the hypocycloid is entirely to the left of the origin and $W_c = 0$. This is the case in Fig. 3A. Because the winding number of $p_c(\Gamma _c)$ is 0, $p_c(z)$ has the minimum number of zeros, 5, as indicated by Example 1.2.

Figure 3B uses $\rho = \frac{14}{9}$ so that $\rho _2< \frac{14}{9} < \rho _3$. The hypocycloid has grown enough to intersect the positive real axis twice. Now $W_c = -2$. Example 1.2 showed 9 zeros for this value of $\rho$ which corresponds to adding $2\left|W_c \right|$ to the minimum number of zeros.

The final image (C) has four intersections with the positive real axis yielding the minimum winding number $W_c = -4$. We know that $p_c(z)$ produces 13 zeros for $c = \frac{16}{9}e^{\frac{i\pi }{3}}$, which is $5+2\left|-4 \right|$ as expected.

Recall that Example 1.2 shows five distinct numbers of zeros. Figure 3 only shows the hypocycloid for three different values of $\rho$, but one can graph $p_c(\Gamma _c)$ for the other two $\rho$ values used in Example 1.2. The resulting hypocycloids will have $W_c = -1$ and $W_c = -3$ and correspond with the remaining possible numbers of zeros.

The next two lemmas are key to determining the winding number of $p_c(\Gamma _c)$.

Lemma 3.6

The hypocycloid $p_c(\Gamma _c)$ intersects the real axis to the right of $-1$ exactly k times.

Proof

It suffices to consider intersections of the translated hypocycloid $p_c(\Gamma _c) + 1$ with the positive real axis.

By part (c) of Lemma 2.5 in [1], a position zero hypocycloid centered at the origin intersects the positive real axis exactly k times. We briefly recall the argument. Let cusp 0 be the cusp on the positive real axis and move counterclockwise, labeling subsequent cusps $1,2,\ldots , n+k-1$. Let $\gamma (r,r')$ denote the arc of the hypocycloid going from r to $r'$. We consider labels modulo $n+k$, so we may write $r'$ as either $r+n$ or $r-k$. The authors of [1] show that $\gamma (r, r')$ crosses the positive real axis if and only if r is in the open upper half-space and $r'$ is in the open lower half-space and prove that this occurs exactly when $1\le r \le k-1$. Of course, an additional intersection occurs for the cusp lying on the positive real axis itself, totalling k intersections.

If the hypocycloid is rotated with a cusp still landing on the positive real axis, the argument from [1] holds just as well. If it is rotated without a cusp landing on the positive real axis, label the cusp immediately beneath this axis as cusp 0, and the cusp immediately above as cusp 1. Then, cusps $1, 2, \ldots , k-1$ are a smaller angle measure from the positive real axis than they were with the position zero hypocycloid, so each arc emanating from these goes from the open upper half-space to the open lower half-space and thus crosses the positive real axis. Finally, the arc $\gamma (k, 0)$ crosses the positive real axis, since k is in the open upper half-space and 0 is in the open lower half-space. We can also see that for $r \ge k+1$ and r in the open upper half-space, $r'$ fails to be in the lower half-space and thus no other arcs cross the positive real axis. So there are k intersections in each case. $\square$

Lemma 3.7

If a position zero hypocycloid intersects itself at a point $P = x + iy$, then $\arg (P + 1) = \frac{\ell \pi }{n+k} + 2\pi j$ for some $\ell ,j \in {\mathbb {Z}}$.

Proof

We again consider the translated hypocycloid $p_c(\Gamma _c)+1$. We label cusps and arcs as above, with cusp 0 on the positive real axis.

Suppose two cusps cross each other. By rotating the figure if necessary, we may assume without loss of generality that the arcs are $\gamma (k,0)$ and $\gamma (r,r-k)$ for $1\le r \le k-1$. Construct two triangles, the first with vertices at the origin and cusp points 0 and k, and the second with vertices at the origin and cusp points $r-k$ and r. These triangles are congruent isosceles triangles. We are particularly interested in the point E of intersection. See Fig. 4.

Fig. 4

Triangles described in the proof of Lemma 3.7

One can argue that all triangles that appear to be congruent in fact are. Thus, $m \angle BOE = m \angle COE$. Furthermore, because each cusp is along a ray making an angle of $\frac{2\ell \pi }{n+k}$ with the positive real axis for some $\ell \in {\mathbb {Z}}$,

$$\begin{aligned} m \angle BOE+m \angle COE=\frac{2\ell \pi }{n+k}, \text{ which } \text{ implies } m \angle BOE = \frac{\ell \pi }{n+k}. \end{aligned}$$

It remains to argue that the arcs intersect on the line through O and E. Indeed, because the triangles AOE and DOE are congruent and the arcs of the hypocycloid are congruent, the arcs must intersect the line through O and E at the same point. The lemma follows. $\square$

We are now ready to compute the winding numbers $W_c$. Recall that n, k are given in the definition of $p_c$ in (1.1).

Lemma 3.8

Let $N_\tau$ be defined as in Eq. (1.2). The hypocycloid $p_c(\Gamma _c)$ has $N_\tau$ distinct intersections with the real axis to the right of its center. These intersections, in turn, correspond to the $N_\tau$ critical $\rho$-values $\rho _j(\tau )$, with $0< \rho _1(\tau )< \dots < \rho _{N_\tau }(\tau )$, at which $p_c(\Gamma _c)$ intersects the origin. The winding number, $W_c$, of $p_c(\Gamma _c)$ around the origin is as follows:

• (a) If $0<\rho <\rho _1(\tau )$, then $W_c = 0$.

• (b) If $\rho _j(\tau )< \rho <\rho _{j+1}(\tau )$ for $1 \le j \le N_\tau -1$, then

  $$\begin{aligned} W_c= {\left\{ \begin{array}{ll} -2j+1 &{} \text {if}\ \tau = \frac{m \pi }{n}\ \text {with}\ m\ \text {even,} \\ -2j &{} \text {if}\ \tau = \frac{m \pi }{n}\ with\ m\ \text {odd,} \\ -j &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

  (3.8)

• (c) If $\rho >\rho _{N_\tau }(\tau )$, then $W_c = -k$.

Proof

We first prove parts (a) and (c); then, we prove (b) and show that the values for $N_\tau$ hold. Note that the results of (b) for m even, and for (a) and (c) are similar to the results in [1]. To compute the winding number $W_c$, we use the crossing condition as stated in [8]. By that condition, we only need to consider a path from the origin going to the right and count how many curves we must cross to “escape" the hypocycloid. Each curve that we cross that is moving from our right to left as we travel away from the origin contributes $-1$ to $W_c$.

Fix the argument of c to be $\tau$ so that $c = \rho e^{i \tau }$ for some $\rho$ and let $\rho$ be arbitrary.

For part (a), recall that the furthest point on the hypocycloid from its center increases in distance from the center as $\rho$ increases. Then, $\rho$-values less than some critical $\rho _1(\tau )$ produce a hypocycloid whose intersections with the real axis all occur to the left of the origin. Then, $W_c=0$ for $0\le \rho < \rho _1(\tau )$.

For part (c), Lemma 3.6 shows that the hypocycloid $p_c(\Gamma _c)$ intersects the real axis to the right of $-1$ exactly k times. Then for fixed $\tau$, as $\rho$ increases, the arcs making the intersections will each move to the right of the origin, eventually producing k total arcs to cross along a path from the origin to the right. Then, there is a value $\rho _{N_\tau }(\tau )$ so that whenever $\rho > \rho _{N_\tau } (\tau )$, all k intersections are to the right of the origin, each contributing $-1$ to $W_c$, giving $W_c = -k$.

For part (b), we treat the three cases of $\tau$ separately.

Case 1 Suppose that $\tau = \frac{m \pi }{n}$ with m even. The hypocycloid $p_c(\Gamma _c)$ has $n+k$ cusps that are an angle of $\frac{2\pi }{n+k}$ apart. By Lemma 3.4, having nonzero $\tau$ rotates the position zero hypocycloid through an angle $\alpha = \frac{\tau n}{n+k}$. Because $\tau = \frac{m \pi }{n}$ with m even, the angle $\alpha$ can be written as $\frac{2\pi m'}{n+k}$, where $m' = \frac{m}{2}$ is an integer. This is then an integer multiple of the angle between two cusps, so the resulting hypocycloid has a cusp on the real axis to the right of $-1$ and is symmetric with respect to this axis. Consequently, the k intersections to the right of the center may not all be distinct, by Lemma 3.7.

Since we are working with a position zero hypocycloid, as established in [1], we have $N_\tau =\lfloor k/2 \rfloor +1$ distinct intersections for all values of k. Thus, by the same argument given in Lemma 2.5 of that paper, as $\rho$ increases, the winding number $W_c$ changes according to the pattern $0, -1, -3, -5,\dots$, which means $W_c = -2j+1$ if $\rho _j(\tau )< \rho < \rho _{j+1}(\tau )$ for $1 \le j \le N_\tau -1$.

Case 2 Suppose that $\tau = \frac{m \pi }{n}$ with m odd. Thus, the position zero hypocycloid is rotated by an angle of $\frac{m\pi }{n+k}$, which is not an integer multiple of the angle between two cusps but is instead a half-integer multiple. Then, the real axis to the right of $-1$ lies exactly between two cusps, and the hypocycloid is symmetric with respect to the real axis. As with case 1, this means that intersections with this axis may not be distinct, by Lemma 3.7.

We once again enumerate the cusps, now letting cusp 0 be the cusp just below the real axis to the right of the center and continuing counterclockwise. The outermost part of the hypocycloid on the real axis to the right of $-1$ is then the intersection between $\gamma (k,0)$ and $\gamma (1,1-k)$ (labeling modulo $n+k$). For each $1 \le r \le k$, the arc $\gamma (r,r-k)$ crosses the real axis to the right of the center of the hypocycloid. Since the position zero hypocycloid is rotated by a multiple of $\frac{\pi }{n+k}$, we know that a self-intersection of the hypocycloid on the real axis is possible. Suppose that $\gamma (r',r'-k)$, $1 \le r' \le k$, intersects $\gamma (r,r-k)$ on the real axis. The average of arguments of the cusps r and $r'-k$ must be some multiple of $2\pi$. Thus,

$$\begin{aligned} \frac{1}{2} \left( \frac{(2r-1)\pi }{n+k} + \frac{(2(r'-k)-1)\pi }{n+k}\right) = 2\pi s \end{aligned}$$

(3.9)

for some $s\in {\mathbb {Z}}$. Solving for $r'$ gives $r' = k+1-r+2s(n+k)$, so $r' = k+1-r$. Thus, the arcs $\gamma (r, r-k)$ and $\gamma (k+1-r,1-r)$ pass through the same point on the real axis. These two arcs are distinct unless $r=k+1-r$, which can only happen if k is odd and $r = (k+1)/2$. This arc that is its own mirror image must be the leftmost intersection of the hypocycloid with the real axis to the right of the center.

Counting the intersections, we see that for k even, every intersection with the real axis is composed of two arcs, so there are k/2 distinct intersections. For k odd, there are $(k-1)/2 + 1$ distinct intersections since we must include the mirrored arc. Together, this implies that for any k, there are $\lfloor (k+1)/2 \rfloor$ intersections with the real axis to the right of the center of the hypocycloid.

Since an intersection with only one distinct arc can only occur at the intersection point closest to the center of the hypocycloid, we conclude that as $\rho$ increases, the winding number $W_c$ changes according to the pattern $0, -2, -4, \ldots$ (excepting the final intersection, depending on the parity of k). Then, $W_c = -2j$ if $\rho _j(\tau )< \rho < \rho _{j+1}(\tau )$ for $1 \le j \le N_\tau -1$.

Case 3 Suppose that $\tau \ne \frac{m\pi }{n}$ for any $m \in {\mathbb {Z}}$. By Lemma 3.7, intersections between distinct arcs of the hypocycloid only occur on the real axis when $\tau$ is an integer multiple of $\frac{\pi }{n}$. Thus, for this third case, every intersection of the hypocycloid with the real axis is distinct. This means that there are exactly k critical $\rho$-values and $N_\tau =k$. Then, we are only crossing one curve at a time as we move from right to left, and the winding number $W_c$ changes by a pattern of $0, -1, -2, \ldots$, so $W_c = -j$ for $\rho _j(\tau )< \rho < \rho _{j+1}(\tau )$ for $1 \le j \le N_\tau -1$. $\square$

We are now prepared to prove the main theorem.

Proof of Theorem 1.1

Let $\tau$ be fixed. Note that Lemma 3.8 already establishes the existence of $N_\tau$ critical values in each case. By Lemma 3.1 and the Argument Principle for Harmonic Functions, the sum of the orders of the zeros of $p_c$ is always n. Furthermore, by Lemma 3.2, all zeros of $p_c$ have order 1 or $-1$.

For part (a), when $0<\rho <\rho _1(\tau )$, $W_c = 0$ and so there are no zeros in the sense-reversing region. Thus, for these $\rho$-values, $p_c$ has only n zeros.

For part (c), since $W_c = -k$ for any $\tau$ when $\rho > \rho _{N_\tau }(\tau )$, there are k distinct zeros in the sense-reversing region and consequently $n+k$ distinct zeros in the sense-preserving region. We conclude that $p_c$ has $n+2k$ distinct zeros for these values of $\rho$.

For part (b), suppose $\rho _j(\tau )< \rho < \rho _{j+1}(\tau )$ for $1 \le j \le N_\tau -1$. As $W_c$ is defined piecewise in Lemma 3.8, we reference its three pieces as cases (i) through (iii), respectively. In case (i), $W_c = -2j+1$ and by the Argument Principle, the number of zeros in the sense-reversing region is $|W_c| = 2j-1$. Then by Lemma 3.1, the number of zeros in the sense-preserving region is $n + |-2j+1| = n+2j-1$, yielding $n+2(2j-1) = n+4j-2$ total zeros. In case (ii), $W_c = -2j$ and again by the Argument Principle, the number of zeros in the sense-reversing region is $|W_c| = 2j$. Then by Lemma 3.1 again, the number of zeros in the sense-preserving region is $n+|-2j| = n+2j$, yielding $n+2(2j)=n+4j$ total zeros. In case (iii), $W_c = -j$ so the number of zeros in the sense-reversing region is $|W_c|=j$, and the number of zeros in the sense-preserving region is $n+|-j|=n+j$, yielding $n+2j$ total zeros. $\square$

Remark 3.9

For bounds on $\rho _1(\tau )$ and $\rho _{N_\tau }(\tau )$, the same computation performed in [1] with $\rho$ substituted for c yields the same results, again showing that the transition points $\rho _j(\tau )$ in the number of zeros occur in a small range of positive real numbers. Thus for $1 \le j \le N_\tau$

$$\begin{aligned} \frac{n}{n+k}\left( \frac{n+k}{k}\right) ^{\frac{k}{n}} \le \rho _1(\tau )< \rho _2(\tau )< \ldots < \rho _{N_\tau }(\tau ) \le \frac{n}{n-k}\left( \frac{n-k}{k}\right) ^{\frac{k}{n}}. \end{aligned}$$

These critical values depend on $\tau$, and one can prove that the bound for $\rho _1(\tau )$ is sharp when $\tau = \frac{m \pi }{n}$ with m even (as in part (i) of case (b) in Theorem 1.1), as this is the case studied in [1] where they determine that this bound is sharp. It appears that the sharpness of the bound for $\rho _{N_\tau }(\tau )$ depends on the parity of k, such that sharpness occurs for $\tau = \frac{m \pi }{n}$ with m the same parity as k.

4 Areas for Further Investigation

• 1. Can we say anything more precise about the critical values $\rho _{j} (\tau )$ for $1 \le j \le N_\tau$?

• 2. Complex-valued harmonic polynomials of the form $p(z)=h(z)-{\overline{z}}$ are related to gravitational lensing. It has been shown in [6] that if the degree of h is $n>1$, then the number of zeros is bounded by $3n-2$. Could the approach of this paper be used to improve that bound or to show that it is sharp?

• 3. What can be proved about the number of zeros of other families of complex-valued harmonic polynomials?