Remark on the roots of generalized Lens equations

TL;DR

This paper investigates the roots of generalized Lens polynomials and demonstrates that the maximum number of roots can be achieved with a specific form of these polynomials, refining previous results using bifurcation techniques.

Contribution

It shows that the maximum number of roots for generalized Lens polynomials can be attained with a modified bifurcation family, extending prior work on harmonic splitting Lens polynomials.

Findings

Maximum roots for generalized Lens polynomials can be achieved with a specific construction.

A slight modification of bifurcation families suffices to reach the maximum number of roots.

The result extends previous bounds on roots of harmonic Lens type polynomials.

Abstract

We consider roots of a generalized Lens polynomial $L(z,\bar z)={\bar z}^m q(z)-p(z)$ and also harmonically splitting Lens type polynomial $L^{hs}(z,\bar z)=r(\bar z)q(z)-p(z)$ and with ${\rm deg}\,q(z)=n$, ${\rm deg}\,r(\bar z)=m$ and ${\rm deg}\,p(z)\le n$. We have shown that there exists a harmonically splitting polynomial $r(\bar z)q(z)-p(z)$ which takes $5n+m-6$ roots, using a bifurcation family of polynomials. In this note, we show that this number can be taken by a generalized Lens polynomial ${\bar z}^mq(z)-p(z)$ after a slight modification of the bifurcation family of a Rhie polynomial.

Equations

$$\begin{array}{rl}f(z,\bar{z})& =f_d(z,\bar{z})+f_{d-1}(z,\bar{z})+\cdots +f_0(z,\bar{z}).\end{array}$$

$$\begin{array}{rl}& {\mathrm{deg}}_{z}f:=\max \{\nu |a_{\nu ,\mu }\ne 0\}\\ & {\mathrm{deg}}_{\bar{z}}f:=\max \{\mu |a_{\nu ,\mu }\ne 0\}\\ & \mathrm{deg}f:=\max \{\mu +\nu |a_{\nu ,\mu }\ne 0\}\end{array}$$

$$L(n+m;n,m)$$

$$L^{hs}(n+m;n,m)$$

$$M(n+m;n,m)$$

$$L(n+m;n,m)\subset L^{hs}(n+m;n,m)\subset M(n+m;n,m).$$

$${\bar{z}}^m-\frac{p(z)}{q(z)},r(\bar{z})-\frac{p(z)}{q(z)}.$$

$$L(z,\bar{z})=\bar{z}-\sum _{i=1}^n\frac{{\sigma }_i}{z-{\alpha }_i}=0,{\sigma }_i,{\alpha }_i\in {\mathbb{C}}^{\ast }.$$

$$\tilde{L}(z,\bar{z}):=L(z,\bar{z})\prod _{i=1}^n(z-{\alpha }_i)\in M(n+1;n,1).$$

$${\ell }_n(z,\bar{z}):=\bar{z}q(z)-p(z),{\mathrm{deg}}_{z}q=n,{\mathrm{deg}}_{z}p\le n,n\ge 2.$$

$${\varphi }_t(z,\bar{z}):=\frac{((t\bar{z}+\gamma {)}^m-{\gamma }^m)}{{\gamma }^{m-1}mt}q(z)-p(z),t\in \mathbb{C}.$$

$$(t\bar{z}+\gamma {)}^{m}q(z)={\gamma }^{m-1}mtp(z)+{\gamma }^{m}q(z).$$

$${\tilde{\varphi }}_t(u):=\frac{(t+\gamma \bar{u}{)}^m-{\gamma }^m{\bar{u}}^m}{{\gamma }^{m-1}mt}\tilde{q}(u)-{\bar{u}}^m\tilde{p}(u)$$

$$\begin{array}{rl}\tilde{q}(u)& =\gamma +\sum _{i=1}^n{b}_i{u}^i\end{array}$$

$${\tilde{\varphi }}_{t,1}(v)$$

$$=$$

$$\Delta :=\{u||u|\le 1/R\},W:=\{u\in \Delta ||v-{\beta }_j|\ge \delta ,j=1,\dots ,m-1\}.$$

$$\begin{array}{r}\left|\frac{(1+\bar{v}{)}^m-{\bar{v}}^m}{{\gamma }^{m-1}m}\tilde{q}(vt)\right|\ge M^{\prime }\delta ,v\in W\end{array}$$

$$|{\varphi }_{t,1}(u)|\ge |t{|}^{m-1}{M}^{\prime }\delta ,u\in W.$$

$$|{\varphi }_{t,2}(u)|\le M^{\prime \prime }|t{|}^m.$$

$$|{\tilde{\psi }}_t(v)|\ge \frac{M^{\prime }}{2}|t{|}^{m-1},\text{for}v\in W$$

$$f(z,\bar{z})={\bar{z}}^m{z}^n-1.$$

$$f(z,\bar{z})={\left(\frac{\bar{z}}{100}+1\right)}^3(z^3-\frac{1}{8})-z^3-\frac{3z^2}{100}+\frac{12513}{100000}.$$

$${\ell }_3(z,\bar{z})=\frac{3}{100}\bar{z}(z^3-\frac{1}{8})-z^3-\frac{3z^2}{100}+\frac{12513}{100000}=0.$$

$$w_1\approx -150+86.6i,w_2=\overline{w_1}.$$

Full text

1. Introduction

Consider a mixed polynomial of one variable $f(z,\bar{z})=\sum_{\nu,\mu}a_{\nu,\mu}z^{\nu}{\bar{z}}^{\mu}$. We consider the number of roots of $f=0$. Assume that $z=\alpha$ is an isolated zero of $f=0$. Put $f(z,\bar{z})=g(x,y)+ih(x,y)$ with $z=x+iy$ where $g=\Re(f)$ and $h=\Im(f)$. We call $\alpha$ a positive simple root (respectively a negative simple root), if the Jacobian $J(g,h)$ is positive (resp. negative) at $z=\alpha$.

1.1. Number of roots with sign

Let $f(z,\bar{z})$ be a given mixed polynomial of one variable, we consider the filtration by the degree:

[TABLE]

Here $f_{\ell}(z,\bar{z}):=\sum_{\nu+\mu=\ell}c_{\nu,\mu}z^{\nu}\bar{z}^{\mu}.$ We consider the case $f_{d}(z,\bar{z})=z^{n}{\bar{z}}^{m}$ with $n+m=d$. The total number of roots of $f(z,\bar{z})=0$ with sign is denoted by $\beta(f)$. Under the above assumption, $\beta(f)=n-m$ by Theorem 20, [5].

1.2. Number of roots without the sign

We assume that roots of $f(z,\bar{z})=0$ are all simple. The number of roots without considering the sign is denoted by $\rho(f)$. Note that $\rho(f)$ is not described by the highest degree part $f_{d}$, which was the case for $\beta(f)$. Consider a mixed polynomial $f(z,\bar{z})=\sum_{\nu,\mu}a_{\nu,\mu}z^{\nu}{\bar{z}}^{\mu}$. We use the definitions

[TABLE]

$\deg_{z}\,f,\,\deg_{\bar{z}}\,f,\,\deg\,f$ are called the holomorphic degree , the anti-holomorphic degree and the mixed degree of $f$ respectively. We consider the following subclasses of mixed polynomials:

[TABLE]

where $p(z),q(z)\in\mathbb{C}[z],r(\bar{z})\in\mathbb{C}[\bar{z}]$. Here $z$ is an affine coordinate of $\mathbb{C}$ but we do not fix $z$. So, a mixed polynomial $f(u,\bar{u})$ is called a generalized Lens polynomial or a harmonically splitting Lens type polynomial if $f$ takes the above form under some affine coordinate $u=z+c$. We have canonical inclusions:

[TABLE]

The class $L(n+m;n,m),L^{hs}(n+m;n,m)$ corresponds to the numerators of harmonic functions

[TABLE]

In particular, $L(n+1;n,1)$ corresponds to the lens equation. We call $\bar{z}^{m}q(z)-p(z)$ a generalized lens polynomial and and $r(\bar{z})q(z)-p(z)$ a harmonically splitting lens type polynomial respectively.

1.3. Lens equation

The following equation is known as the lens equation.

[TABLE]

We identify the left side rational function with the mixed polynomial given by its numerator

[TABLE]

Theorem 1**.**

(Khavinson-Neumann [2]) The number of roots of $L$ or $\tilde{L}$ is bounded by $5n-5$ for $n\geq 2$.

Rhie gave an explicit polynomial which takes this bound $5n-5$ in [7]. Thus this bound is optimal. On the other hand, $\rho(L)\equiv n-1$ mod $2$ by Theorem 20, [5].

Theorem 2**.**

(P. Bleher, Y. Homma, L. Ji and P. Roeder [1] )* The set of possible values of $\rho(f)$ for $f\in L(n+1,n,1)$ is equal to $\{n-1,n+1,\cdots,5n-5\}$.*

1.4. Bifurcation family

In [3], we have constructed a generalized Lens type polynomial which take $5(n-m)$-roots if $n>3m$ and we have asked if this is an optimal upper bound or not. On the other hand, for the space of harmonically splitting Lens type polynomials $L^{hs}(n+m;n,m)$, we studied a bifurcation family $\psi_{t}(z,\bar{z}):=t{\bar{z}}^{m}+\ell_{n}(z,\bar{z})\in L^{hs}(n+m;n,m)$ starting from a given Lens polynomial $\ell_{n}(z,\bar{z})$ with $\rho(\ell_{n})=k$. Let $\alpha_{1},\dots,\alpha_{k}$ be the roots of $\ell_{n}$. We have proved

Theorem 3**.**

([3])* $\psi_{t}=0$ has exactly $k+m-1$ roots for small $t$. Furthermore $k$ roots of them are near each $\alpha_{j}$ with the same sign and $m-1$ roots are newly born roots bifurcated from $z=\infty$. These new roots are negative roots.*

2. Main result

2.1. Modification of the bifurcation family and the main result

In this note, we answer the above question negatively. In fact, we modify the above bifurcation family to prove the same assertion for generalized Lens polynomials. We start from an arbitrary Lens type polynomial with only simple roots:

[TABLE]

Put $k=\rho(\ell_{n})$ and let $\alpha_{1},\dots,\alpha_{k}$ be the roots of $\ell_{n}$. Note that $n-1\leq k\leq 5n-5$ and $k\equiv n-1$ mod $2$. Put $\gamma$ be the coefficient of $z^{n}$ in $q(z)$. $\gamma$ is non-zero. Consider its small purturbation of $\ell_{n}(z)$ in the space of generalized Lens polynomials $L(n+m;n,m)$:

[TABLE]

Note that $\phi_{0}(z,\bar{z})=\ell_{n}(z,\bar{z})$ and $\phi_{t},\,t\neq 0$, corresponds to the generalized Lens equation

[TABLE]

In fact, by the change of coordinate $u=\bar{t}z+\bar{\gamma}$, $\phi_{t}$ takes the expected form.

Theorem 4**.**

For sufficiently small $t\in\mathbb{C},\,|t|\ll 1$, $\rho(\phi_{t})=k+m-1$. Furthermore

1. (1)

$k$* roots $\alpha_{j}(t),j=1,\dots,k$ are small deformation of $\alpha_{j},\,j=1,\dots,k$ and the sign of $\alpha_{j}(t)$ is the same as that of $\alpha_{j}$.* 2. (2)

$m-1$* new roots $\beta_{a}(t),a=1,\dots,m-1$ are born at infinity i.e., $\beta_{a}(0)=\infty$ and they are negative roots.*

Taking $\ell_{n}(z,\bar{z})$ to be a Rhie’s polynomial, we get $\rho(\phi_{t})=5n+m-6$.

Proof.

For sufficiently small $t$ and for each root $\alpha$ of $\ell_{n}$, by the continuity of the roots, there exists a root $\alpha(t)$ of $\phi_{t}=0$ in a neighborhood of $\alpha$ with $\alpha(0)=\alpha$ and $\alpha(t)$ has the same orientation as $\alpha$. For $t\neq 0$, we know that $\beta(\phi_{t})=n-m$ for $t\neq 0$ and $\beta(\ell_{n})=n-1$. Thus it is clear that we need at least $m-1$ negative roots. Take a large $R>0$ so that $|\alpha_{j}|\leq R/2$ for any $j=1,\dots,k$. For any small $\varepsilon>0$, there exists $\delta(\varepsilon)>0$ such that $\phi_{t}$ has $k$ roots near each $\alpha_{j}(t),\,|t|\leq\delta(\varepsilon)$ with the same sign as $\alpha_{j}$ in the original equation $\ell_{n}=0$. We may assume that $|\alpha_{j}(t)-\alpha_{j}|\leq\varepsilon$ for $j=1,\dots,k$ and there are no other roots of $\phi_{t}(z)=0$ in the disk $D_{R}=\{z\,|\,|z|\leq R\}$. On the other hand, as $\beta(\phi_{t})=n-m,\,t\neq 0$, we have the property $n-m\equiv k-(m-1)$, mod $2$. Thus $\phi_{t}$ has at least $m-1$ new negative roots outside of the disk $D_{R}$.

We assert that $\phi_{t}$ obtains exactly $m-1$ new negative roots near infinity. To see this, we change the coordinate $u=1/z$ and dividing (2) by ${\bar{z}}^{m}z^{n}$, we get

[TABLE]

where $\tilde{q},\tilde{p}$ are polynomials defined as $\tilde{q}(u)=u^{n}q(1/u),\,\tilde{p}(u)=u^{n}p(1/u)$. By the asumption $\deg\,q(z)=n$, we can write

[TABLE]

We will show that for a sufficiently small $t>0$, there exist exactly $m-1$ roots $u(t)$ which converges to 0 as $t\to 0$. Put $\tilde{\phi}_{t,1},\tilde{\phi}_{t,2}$ be the first and the second term of (3). Putting $u=vt$ for $t\neq 0$, we can write $\tilde{\phi}_{t,1}$ as

[TABLE]

where $h(v)$ is a polynomial with a non-zero constant. $h(v)=0$ has $m-1$ simple roots, and we put them as $v=\beta_{1},\dots,\beta_{m-1}$. Consider the disk at infinity and its subset $W$:

[TABLE]

On $\Delta$, we estimate $1/M\leq|\tilde{q}(vt)|\leq M$ for some $M>0$. Taking a small number $\delta>0$, we can make

[TABLE]

with some constant $M^{\prime}>0$. Or equivalently,

[TABLE]

Taking $t$ small, we can make the second term of (3) as small as possible on $\Delta$ comparing with $|t|^{m-1}$. More precisely, there exists a positive number $M^{\prime\prime}$ such that

[TABLE]

Thus if $|t|$ is sufficiently small,

[TABLE]

which implies $\tilde{\psi}_{t}(v)=0$ has one simple negative root in $D_{j}:=\{v\,|\,|v-\beta_{j}|\leq\delta\}$ for $j=1,\dots,j$ and no root on $W$. The negativity of these $m-1$ new roots is clear as $\beta(\phi_{t})=n-m$ and $\beta(\ell_{n})=n-1$. This completes the proof. ∎

2.2. Possible values of $\rho$

Assume that $n\geq m$. Combining Theorem 2, we can see that

Corollary 5**.**

$\rho(f)$* for $f\in L(n+m;n,m)$ can takes the values $\{n+m-2,\dots,5n+m-6\}$.*

As for the lower values $\{n-m,\dots,n+m-4\}$, we know that these values can be taken by some polynomials in $M(n+m;n,m)$. We do not know if these values can be taken in $L(n+m;n,m)$ or $L^{hs}(n+m;n,m)$ except $n-m$. For $n-m$, it can be taken by

[TABLE]

2.3. Example

Consider

[TABLE]

This is a bifurcation of a Rhie type polynomial $\ell_{3}(z,\bar{z})$ with $\rho(\ell_{3})=10$ where

[TABLE]

Then $\Re f=0$ is the green curve and $\Im f=0$ is the union of the real axis and the red curve. The root $f=0$ is the intersection of two curves and we see 10 roots in the graph. Actually there are two roots $w_{1},w_{2}$ which are big :

[TABLE]

and thus $\rho(f)=12$.