# Dynamic State Estimation-Based Protection
for Induction Motor Loads

Arthur K. Barnes 1*  —, Sarbajit Basu 2*  —, and Adam Mate 1*  — Manuscript
submitted: Jul. 15, 2022. Current version: Aug. 15, 2022. 1 The authors are
with the Advanced Network Science Initiative at Los Alamos National
Laboratory, Los Alamos, NM 87545 USA.
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>The author is with the Klipsch
School of Electrical and Computer Engineering at the New Mexico State
University, Las Cruces, NM 88003 USA. Email<EMAIL_ADDRESS>Approved for public release; distribution is
unlimited.978-1-6654-9921-7/22/$31.00 ©2022 IEEE

###### Abstract

Ensuring protective device coordination is critical to maintain the resilience
and improve the reliability of large microgrids. Inverter-interfaced
generation, however, poses significant challenges when designing protection
systems. Traditional time-overcurrent protective devices are unsuitable on
account of the lack of fault current. Present industry practice is to force
all inverters to shut down during faults, which prevents large microgrids from
operating in a resilient and reliable manner. Dynamic state estimation (DSE)
has been proposed for line protection, and more recently for the protection of
load buses or downstream radial portions of microgrids. However, only passive
loads with series resistive-inductive loads have been tested with DSE, even
though the behavior of dynamic loads – such as induction motors or power
electronics – may differ significantly during faults. This paper considers the
case of applying DSE to protecting a load bus serving a three-phase induction
motor.

###### Index Terms:

power system operation, microgrid, distribution network, protection, dynamic
state estimation.

##  I Introduction

Microgrids have been a key tool to integrate distributed renewable energy
resources into the bulk energy system. When operating under islanded
conditions, microgrids are reliant upon not only inverter run energy storage,
but renewable resources as well. Although they are reasonably reliable,
economical, and low greenhouse gas contributors, inverter fed resources –
commonly photovoltaic panels and wind farms – suffer from issues such as
availability – variability in the levels of irradiation and wind available
during the day – that makes their power output variable and limited system
inertia [1]. Under faulted conditions, the current drawn by the system when
fed by a synchronous machine is significantly higher and is thus ideal for an
overcurrent relay to provide protection. However, an inverter is limited at
the amount of current it can provide, therefore limiting the ability to use
conventional devices to provide protection; also, the response of an inverter
fed device to a change is fast, and thus control devices must respond quickly
as well [2].

Electrical loads today are dominated by power electronic converter run
devices, and single-phase and three-phase induction machines. Most devices
have an inverter involved for powering them, while induction motors – the
mainstay of HVAC systems, agricultural systems, and several industrial
applications owing to the low cost of construction, easy maintenance, and high
efficiency – require protection mechanisms in place for safe and reliable
operation [3, 4]. Induction motors, owing to their non-linear behavior,
undergo prolonged periods of transiency until they reach steady-state; this
warrants the need to ensure that system parameters during the period of
transition does not reach extremely high values [5]. Under faulted conditions
– be it stator faults, rotor faults, or external line faults – the current
drawn by the machine is significantly higher, which in turn can impact the
health of the device [6]. This becomes more profound in an inverter fed
microgrid, where the system has limited to no inertia – due to the lack of
stored kinetic energy – that could be harvested to adjust to intermittent
loading conditions or fault conditions [7]. Consequently, it becomes important
to study the protection of induction motors in a microgrid.

Protection methods enable the resilient and reliable operation of microgrids
by maintaining protective device coordination to minimize the extent of
outages. Conventional approaches have focused on protecting just the
inverters, resulting in the whole microgrid shutting down during a fault;
additionally, these approaches do not scale to large networks [8]. Distance
relaying techniques have been explored for protecting inverter-interfaced
microgrids: Dewadasa et al. [9] suggested limiting the per phase current in
faulted conditions by using distance relaying; Kar et al. [10] demonstrated
the use of differential relaying based on the discrete S-transform performed
on the change in system current magnitude under fault conditions; and Huang
[11] used an adaptive protection system, in which multiple relays co-ordinate
among themselves to predict the occurrence of a fault and recommend corrective
action. All these, however, employ the use of legacy programmed protection
devices and does not cater to system dynamics or protect the system against
bi-directional power flow. It is unequivocal that new protection methods are
needed to support the deployment of large microgrids.

Settingless protection addresses these shortcomings by employing Dynamic State
Estimation (DSE) to generate system measurements such that protective control
action is determined based on the mismatch between the measurements and
estimates generated using accurate system models [12, 13, 14]. DSE allows for
increased accuracy in tracking system dynamics and better system estimates,
therefore allowing better system operation and control [15]; it can accurately
track system dynamics and can effectively predict the fault characteristics
across microgrids of varying sizes under high penetration of renewable energy
resources [16]. DSE has been used to develop centralized protection schemes
[16, 17] and to detect hidden failures in substation protection systems [18].
More recently, it has been proposed for protection of load buses [19] and
downstream radial sections of microgrids [13, 20]. It is also finding an
increased relevance in power system operation and monitoring applications
[21], owing to the implementation of accurate dynamic models that provide
reliable estimates, improved oscillation monitoring, and increased potential
for decentralized control [15].

This work investigates the application of DSE for load bus protection, while
considering the dynamic behavior of an induction motor. Section II presents
the dynamic simulation model selected for an induction motor, the DSE
formulation derived from the simulation model, and the solution method for the
DSE problem. Section III presents a case study system based on a numerical
simulation of an induction motor under several different fault scenarios.
Section IV presents experimental results from the numerical simulation and DSE
algorithm. Finally, Section V summarizes the conclusions regarding the utility
of DSE for load bus protection.

##  II Methodology

The induction machine model used in this work is a Julia [22] implementation
of the MATLAB asynchronous machine model [23], which has been developed using
the induction motor models proposed by [3] and [24].

###  II-A 5th\- order Induction Motor State Estimation Model

To convert time-varying induction motor inductances into constant values, in
order to simplify the solution of the governing differential equations, the
quantities for the machine inputs and states were converted into a rotating
reference frame, as described below.

Rotating reference-frame transform from [3] (3.3-4):

$\begin{split}&\begin{bmatrix}v_{q}(t)\\\ v_{d}(t)\\\
v_{0}(t)\end{bmatrix}=\\\
&=\frac{2}{3}\begin{bmatrix}\cos(\theta(t))&\cos(\theta(t)-\alpha)&\cos(\theta(t)+\alpha)\\\
\sin(\theta(t))&\sin(\theta(t)-\alpha)&\sin(\theta(t)+\alpha)\\\
1/2&1/2&1/2\end{bmatrix}\begin{bmatrix}v_{a}(t)\\\ v_{b}(t)\\\
v_{c}(t)\end{bmatrix}\end{split}$ (1)

where $\alpha=\frac{2}{3}\pi$; $v_{a}(t)$, $v_{b}(t)$, and $v_{c}(t)$ are the
line-ground voltages on phases A, B and C, respectively; and $v_{q}(t)$,
$v_{d}(t)$, and $v_{0}(t)$ are the voltages on the q, d, and 0 axes,
respectively.

Magnetizing flux linkages from [3] (6.14-15) – (6.14-17):

$\dot{\psi}_{qs}(t)=v_{qs}(t)-\omega\cdot\psi_{ds}(t)-R_{s}\cdot i_{qs}(t)$
(2)

where $v_{qs}(t)$ is the stator voltage in the q-axis; and $\psi_{ds}(t)$ is
the stator flux in the d-axis.

$\dot{\psi}_{ds}(t)=v_{ds}(t)+\omega\cdot\psi_{qs}(t)-R_{s}\cdot i_{ds}(t)$
(3)

where $v_{ds}(t)$ is the stator voltage in the d-axis; and $\psi_{qs}(t)$ is
the stator flux in the q-axis.

$\dot{\psi}_{qr}^{\prime}(t)=v_{qr}^{\prime}(t)-(\omega-\omega_{r})\cdot\psi_{dr}^{\prime}(t)-R_{r}^{\prime}\cdot
i_{qr}^{\prime}(t)$ (4)

where $v_{qr}^{\prime}(t)$ is the rotor voltage referred to the stator in the
q-axis; and $\psi_{dr}^{\prime}(t)$ is the rotor flux in the d-axis referred
to the stator.

$\dot{\psi}_{dr}^{\prime}(t)=v_{dr}^{\prime}(t)+(\omega-\omega_{r})\cdot\psi_{qr}^{\prime}(t)-R_{r}^{\prime}\cdot
i_{dr}^{\prime}(t)$ (5)

where $v_{dr}^{\prime}(t)$ is the rotor voltage referred to the stator in the
d-axis; and $\psi_{qr}^{\prime}(t)$ is the rotor flux in the q-axis referred
to the stator.

In Equations (2) – (5): $\omega(t)$ is the synchronous speed; $R_{s}$ is the
stator resistance; $R^{\prime}_{r}$ is the rotor resistance; and
$\omega_{r}(t)=P\omega(t)$, where $P$ is the number of poles in the induction
machine and $\omega_{r}(t)$ is the angular velocity of the rotor.

Flux linkages expressed as a function of currents, from [3] (6.14-4) –
(6.14-6):

${\psi}_{qs}(t)=L_{s}\cdot i_{qs}(t)+L_{m}\cdot i_{qr}^{\prime}(t)$ (6)
${\psi}_{ds}(t)=L_{s}\cdot i_{ds}(t)+L_{m}\cdot i_{dr}^{\prime}(t)$ (7)
${\psi}_{qs}(t)=L_{r}^{\prime}\cdot i_{qr}^{\prime}(t)+L_{m}\cdot i_{qs}(t)$
(8) ${\psi}_{qs}(t)=L_{r}^{\prime}\cdot i_{qr}^{\prime}(t)+L_{m}\cdot
i_{qs}(t)$ (9)

In Equations (2) – (9): $i_{qs}(t)$ and $i_{ds}(t)$ are the stator currents;
$i_{qr}(t)^{\prime}$ and $i_{dr}(t)^{\prime}$ are the rotor currents referred
to the stator; $L_{s}$ is the stator inductance; $L_{r}^{\prime}$ is the rotor
inductance; and $L_{m}$ is the magnetizing inductance.

Electrical torque from [3] (6.6-16)–(6.6-17):

$\displaystyle T_{e}(t)$
$\displaystyle=\frac{3}{2}\cdot\frac{P}{2}\cdot\frac{1}{\omega}\cdot\left(\psi^{\prime}_{qr}(t)\cdot
i^{\prime}_{dr}(t)-\psi^{\prime}_{dr}(t)\cdot i^{\prime}_{qr}(t)\right)$ (10)
$\displaystyle=\frac{3}{2}\cdot\frac{P}{2}\cdot\frac{L_{m}}{D}\frac{1}{\omega}\cdot\left(\psi_{qs}(t)\cdot\psi^{\prime}_{dr}(t)-\psi^{\prime}_{qr}(t)\cdot\psi_{ds}(t)\right)$
(11)

Rotor speed from [3] (6.3-8):

$\dot{\omega}_{m}(t)=\frac{P}{2J}\cdot\left(T_{e}(t)-F\cdot
w_{m}-T_{m}(t)\right)$ (12)

where $T_{e}$ is the electromagnetic torque; $T_{m}$ is the shaft mechanical
torque; $J$ is the combined rotor and load inertia coefficient; and $F$ is the
combined rotor and load viscous friction coefficient.

###  II-B 5th\- order Induction Motor State Estimation Model

To perform state estimation, the above equations from the simulation model
were reformulated as State-Output Mapping, described in [25].

Observables

$\mathbf{y}=\begin{bmatrix}v_{q}(t)&v_{d}(t)&i_{q}(t)&i_{d}(t)&\\\
z_{\psi_{qs}}(t)&z_{\psi_{ds}}(t)&z_{\psi^{\prime}_{qr}}(t)&z_{\psi^{\prime}_{dr}}(t)&z_{Te}(t)\end{bmatrix}^{T}$
(13)

State

$\mathbf{x}=\begin{bmatrix}\psi_{qs}(t)&\psi_{ds}&\psi^{\prime}_{qr}(t)&\psi^{\prime}_{dr}(t)&T_{e}(t)&\omega_{r}(t)\end{bmatrix}^{T}$
(14)

State-Output Mapping

The algebraic voltage equations:

$\begin{split}v_{qs}(t)=\frac{R_{s}\cdot
L^{\prime}_{lr}}{D}\cdot\psi_{ds}(t)-\omega(t)\cdot\psi_{qs}(t)-\frac{R_{s}\cdot
L_{m}}{D}\cdot\psi^{\prime}_{dr}(t)\end{split}$ (15)
$\begin{split}v_{ds}(t)=\frac{R_{s}\cdot
L^{\prime}_{lr}}{D}\cdot\psi_{qs}(t)+\omega(t)\cdot\psi_{ds}(t)-\frac{R_{s}\cdot
L_{m}}{D}\cdot\psi^{\prime}_{qr}(t)\end{split}$ (16)

Algebraic current equations:

$\begin{split}i_{qs}(t)=\frac{L^{\prime}_{lr}}{D}\cdot\psi_{ds}(t)-\frac{L_{m}}{D}\cdot\psi^{\prime}_{dr}(t)\end{split}$
(17)
$\begin{split}i_{qs}(t)=\frac{L^{\prime}_{lr}}{D}\cdot\psi_{qs}(t)-\frac{L_{m}}{D}\cdot\psi^{\prime}_{qr}(t)\end{split}$
(18)
$\begin{split}i^{\prime}_{qr}(t)=\frac{1}{L_{m}}\cdot\left(\psi_{qs}(t)-L_{s}\cdot
i_{qs}(t)\right)\end{split}$ (19)
$\begin{split}i^{\prime}_{dr}(t)=\frac{1}{L_{m}}\cdot\left(\psi_{ds}(t)-L_{s}\cdot
i_{ds}(t)\right)\end{split}$ (20)

Algebraic electrical torque equation:

$\begin{split}z_{te}(t)=T_{e}(t)-\frac{3}{2}\cdot\frac{PL_{m}}{D}\cdot\left(\psi_{qs}(t)\cdot\psi^{\prime}_{qr}(t)-\psi^{\prime}_{qr}(t)\cdot\psi_{ds}(t)\right)\end{split}$
(21)

In Equations (15) – (21):

$D=L^{\prime}_{lr}\cdot L_{ls}-L_{m}^{2}$ (22)

Differential flux equations:

$\begin{split}z_{\psi_{qs}}(t)=\psi_{qs}(t)-\psi_{qs}(t-\Delta
t)-\int_{t-\Delta t}^{t}\dot{\psi}_{qs}(\tau)\;d\tau\end{split}$ (23)

where

$\begin{split}\dot{\psi}_{qs}(\tau)=v_{qs}(\tau)-\omega\cdot\psi_{ds}(\tau)-R_{s}\cdot
i_{qs}(\tau)\end{split}$
$\begin{split}z_{\psi_{ds}}(t)=\psi_{ds}(t)-\psi_{ds}(t-\Delta
t)-\int_{t-\Delta t}^{t}\dot{\psi}_{ds}(\tau)\;d\tau\end{split}$ (24)

where

$\begin{split}\dot{\psi}_{ds}(\tau)=v_{ds}(\tau)+\omega\cdot\psi_{qs}(\tau)-R_{s}\cdot
i_{ds}(\tau)\end{split}$
$\begin{split}z_{\psi^{\prime}_{qr}}(t)=\psi^{\prime}_{qr}(t)-\psi_{qr}(t-\Delta
t)-\int_{t-\Delta t}^{t}\dot{\psi}^{\prime}_{qr}(\tau)\;d\tau\end{split}$ (25)

where

$\begin{split}\dot{\psi}^{\prime}_{qr}(\tau)=v^{\prime}_{qr}(\tau)-\left(\omega(\tau)-P\cdot\omega_{r}(\tau)\right)\cdot\psi^{\prime}_{dr}(\tau)-R_{rp}\cdot
i^{\prime}_{qr}(\tau)\end{split}$
$\begin{split}z_{\psi^{\prime}_{dr}}(t)=\psi^{\prime}_{dr}(t)-\psi_{dr}(t-\Delta
t)-\int_{t-\Delta t}^{t}\dot{\psi}^{\prime}_{dr}(\tau)\;d\tau\end{split}$ (26)

where

$\begin{split}\dot{\psi}^{\prime}_{dr}(\tau)=v^{\prime}_{dr}(\tau)-\left(\omega(\tau)+P\cdot\omega_{r}(\tau)\right)\cdot\psi^{\prime}_{qr}(\tau)-R_{rp}\cdot
i^{\prime}_{dr}(\tau)\end{split}$

Differential rotor speed equation:

$\begin{split}z_{\omega_{r}}(t)=\omega_{r}(t)-\omega_{r}(t-\Delta
t)-\int_{t-\Delta t}^{t}\dot{\omega}_{r}(\tau)\;d\tau\end{split}$ (27)

where

$\begin{split}\dot{\omega}_{r}(\tau)=\frac{P}{2J}\cdot\left(T_{e}(\tau)-F\cdot\omega_{r}(\tau)-T_{m}(\tau)\right)\end{split}$

Discretized Observables

$\mathbf{y}=\begin{bmatrix}\mathbf{v}_{d}&\mathbf{v}_{q}&\mathbf{i}_{d}&\mathbf{i}_{q}&\\\
\mathbf{z}_{\psi_{qs}}&\mathbf{z}_{\psi_{ds}}&\mathbf{z}_{\psi^{\prime}_{qr}}&\mathbf{z}_{\psi^{\prime}_{dr}}&\mathbf{z}_{Te}\end{bmatrix}^{T}$
(28)

Discretized State

$\mathbf{x}=\begin{bmatrix}\mathbf{\psi}_{qs}&\psi_{ds}&\mathbf{\psi}^{\prime}_{qr}&\mathbf{\psi}^{\prime}_{dr}&\mathbf{T}_{e}&\mathbf{\omega}_{r}\end{bmatrix}^{T}$
(29)

To perform dynamic state estimation, which was formulated as a discrete-time
problem, it is necessary to discretize the governing equations for the
induction motor. The discretization of the algebraic equations is trivial,
while the discretization of the above differential equations was performed
with the trapezoidal rule:

$f[i]-f[i-1]\approx\frac{1}{\Delta t}\int_{t-\Delta t}^{t}f(\tau)d\tau$ (30)

###  II-C Solution of the State Estimation Model

Given the discretized state-output relationship $h(n)$, the Jacobian $H(n,n)$
can be calculated either analytically – based on the discretized algebraic and
diffential state equations – or numerically. While for a real-time
implementation an algebraic representation of the Jacobian is required, for a
proof-of-concept – such as that illustrated here – a numerical Jacobian can be
calculated. For this study, the Julia library FiniteDiff.jl [26] is used for
numerical Jacobian calculation.

Given $h(\cdot)$ and $\mathbf{H}$, the state of the system can be solved by
the following updated equations:

$\epsilon_{i}=y-h(x_{i})$ (31)
$x_{i+1}=x_{i}+(\mathbf{H}_{i}^{T}\cdot\mathbf{H}_{i})^{-1}\mathbf{H}_{i}^{T}\cdot\epsilon_{i}$
(32)

This process is repeated iteratively until either the maximum number of
iterations is reached or the algorithm has converged, indicated by the change
in the log of the squared error falling below a specified threshold:

$\Delta
J_{i}=|\log|\epsilon_{i}^{*}\cdot\epsilon_{i}|-\log|\epsilon_{i-1}^{*}\cdot\epsilon_{i-1}||$
(33)

The measurement error test is performed as follows:

$p=F_{m-n}(J_{i})\geq 0.95$ (34)

where $F_{m-n}$ is the Chi-squared cumulative distribution function for $m-n$
degrees of freedom, in which $m-n$ is the number of linearly independent
observables.

The final algorithm is illustrated below:

1 initialize state vector $x$ to small normally-distributed random numbers

2 while  _change in error is greater than threshold or maximum iteration limit
is reached_ do

3 calculate the estimated output $y_{est}$ from h(x)

4 calculate the sum-squared error $J$ between measured output $y$ and
$y_{est}$

5 calculate the change in the state variable $x$ based on the update equation

6

7 end while

8calculate chi-squared CDF based on the final value of $J$

Algorithm 1 DSE Algorithm Pseudocode

##  III Numerical Experiments

The DSE algorithm was implemented in Julia 1.7 64-bit, on a computer with an
Intel Xeon® E-2276M 2.8 GHz CPU with 128 GB RAM, running MS Windows 10 21H2.

The DSE algorithm was applied to a small single-bus case study system,
illustrated in Fig. 1. It was implemented in MATLAB Simulink® in the
Specialized Power Systems blockset. The system consists of a 3.75 kW (5 hp)
460 V induction motor, supplied by a voltage source, with parameters
illustrated in Table I. The motor is direct-online started and drives a
constant-torque load.

Figure 1: MATLAB Simulink® SimScape model for an induction motor with a fault at the terminals. TABLE I: Case Study Parameters Subsystem | Parameter | Symbol | Value | Units
---|---|---|---|---
| Nominal Power | $P$ | 3.73 | kW
| Voltage (line-line) | $V_{ll}$ | 460 | V
| Frequency | $f$ | 60 | Hz
| Pole Pairs | $P$ | 2 |
| Inertia | $J$ | 0.02 | $\mathrm{kg\cdot m^{2}}$
Stator | Resistance | $R_{s}$ | 1.115 | $\Omega$
Stator | Inductance | $L_{ls}$ | 5.974 | mH
Rotor | Resistance | $R_{r}^{\prime}$ | 1.083 | $\Omega$
Rotor | Inductance | $L_{lr}^{\prime}$ | 5.974 | mH
Rotor | Mutual Inductance | $L_{m}$ | 203.7 | mH
Load | Torque | $T_{m}$ | 50 | $\mathrm{N\cdot m}$
Fault | Fault Resistance | $R_{f}$ | 5.0 | $\Omega$
Fault | Ground Resistance | $R_{g}$ | 0.1 | $\Omega$

The sequence of events in the simulation is as follows:

  1. 1.

the motor is connected to power at $0~{}sec$,

  2. 2.

the constant-torque load is activated at $3~{}sec$,

  3. 3.

a fault is applied at $5~{}sec$, and

  4. 4.

the fault is cleared at $5.25~{}sec$.

The following quantities are recorded during the simulation:

  * •

line-ground voltage on each phase at the voltage source,

  * •

current on each phase at the voltage source,

  * •

mechanical torque drawn by the load, and

  * •

rotor shaft speed.

For the voltage and current measurements, Park’s transformation was performed
to produce the discretized set of observables in (28).

Four different scenarios were considered (Table II). The transformed
measurements were down-sampled to a 100 Hz sample rate and applied to the DSE
algorithm during the fault period.

TABLE II: Case Study Results Case | Phasing |  $\chi^{2}$ CDF
---|---|---
No Fault | – | 0.988
Line-Ground Fault | AG | 0.925
Line-Line Fault | AB | 0.413
Three-Phase Ground Fault | ABCG | 0.800

##  IV Experiment Results

Measured voltages and currents, for the four fault scenarios considered, are
illustrated in Figs. 2 – 5. Applying the DSE algorithm during the fault period
yields the Chi-Squared statistic in column 3 of Table II. Testing to a 95 %
confidence interval allow for faulted conditions to be detected.

Figure 2: Measured voltage and current for unfaulted conditions. Figure 3:
Measured voltage and current for a line-ground fault. Figure 4: Measured
voltage and current for a line-line fault. Figure 5: Measured voltage and
current for a three-phase-ground fault.

##  V Conclusions

This work investigated the application of DSE for load bus protection with
dynamic loads, in the case of a three-phase induction motor. Results suggest
that DSE is a feasible method for protection under common fault types based on
the ability of the DSE to accurately detect faults on the case study system
with a confidence test.

As DSE is a generalisation of differential protection, it can be applied to a
wide range of power conversion elements: not only to conventional elements
such as transformers, transmission lines, and shunt capacitors, but to
elements such as motor loads in microgrids as well. It is expected that this
protection method can be applied to other forms of dynamic loads, such as
single-phase induction motors or active rectifiers as well. Additionally, it
is expected that this method could be extended to not only model individual
loads, but to be representative of composite load models as well – along the
lines of the WECC CMPLDW model – representing the dynamic behavior of the
downstream radial portions of microgrids.

Future work will aim to investigate these above described other applications,
in addition to the feasibility of applying DSE for load bus protection in a
real-time implementations.

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# Continuity of the roots of a polynomial

Melvyn B. Nathanson  Department of Mathematics
Lehman College (CUNY)
Bronx, NY 10468<EMAIL_ADDRESS>and  David A. Ross
Department of Mathematics
University of Hawaii at Manoa
Honolulu, HI 96822<EMAIL_ADDRESS>
###### Abstract.

Let $K$ be an algebraically closed field with an absolute value. This note
gives a simple proof of the classical result that the roots of a polynomial
with coefficients in $K$ are continuous functions of the coefficients of the
polynomial.

###### Key words and phrases:

Polynomials, roots of polynomials, deformations of polynomial, continuity of
roots, elementary methods.

###### 2010 Mathematics Subject Classification:

12D10, 12E05, 12J10, 12L15

##  1\. Continuity of the roots

Let $K$ be an algebraically closed field with an absolute value (for example,
the field $\mathbf{C}$ of complex numbers or the algebraic closure of the
field $\mathbf{Q}_{p}$ of $p$-adic numbers). We denote by $K[z]$ the ring of
polynomials with coefficients in $K$.

Let $(a_{i})_{i\in I}$ and $(b_{i})_{i\in I}$ be sequences of elements of $K$
and let $\delta>0$. The sequence $(b_{i})_{i\in I}$ is a _$\delta$
-deformation_ of the sequence $(a_{i})_{i\in I}$ if

$|b_{i}-a_{i}|<\delta$

for all $i\in I$.

Let $f(z)=\sum_{i=0}^{n}a_{i}z^{i}$ and $g(z)=\sum_{i=0}^{n}b_{i}z^{i}$ be
polynomials in $K[z]$ of degree $n$. Let $\delta>0$. The polynomial $g(x)$ is
a _$\delta$ -deformation_ of the polynomial $f(z)$ if the sequence
$(b_{0},b_{1},\ldots,b_{n})$ of coefficients of $g(z)$ is a
$\delta$-deformation of the sequence $(a_{0},a_{1},\ldots,a_{n})$ of
coefficients of $f(z)$.

In the algebraically closed field $K$, every polynomial of degree $n$ in
$K[z]$ has $n$ roots, counting multiplicity. Let
$(\zeta_{1},\ldots,\zeta_{n})$ be the sequence of roots of $f(z)$ and let
$(\omega_{1},\ldots,\omega_{n})$ be the sequence of roots of $g(z)$, where a
root of multiplicity $m$ appears $m$ times in the sequence. A classical
theorem on the continuity of roots of a polynomial states that, for every
$\varepsilon>0$, there exists $\delta>0$ such that if the polynomial $g(x)$ is
a $\delta$-deformation of the polynomial $f(x)$, then there is a permutation
$\sigma\in S_{n}$ such that the rearranged sequence of roots
$\left(\omega_{\sigma(1)},\ldots,\omega_{\sigma(n)}\right)$ is an
$\varepsilon$-deformation of $(\zeta_{1},\ldots,\zeta_{n})$. Most proofs of
this result require sophisticated or technically difficult mathematics (cf.
[1]–[8]). The object of this note is to give a proof that uses only high
school algebra and the definition of continuity.

The open ball with center $\zeta\in K$ and radius $\varepsilon>0$ is the set

$B(\zeta,\varepsilon)=\\{z\in K:|z-\zeta|<\varepsilon\\}.$

######  Lemma 1.

Let $\zeta_{1},\ldots,\zeta_{\ell}$ be distinct elements of the field $K$. If

(1) $0<\varepsilon\leq\frac{1}{2}\min\left(|\zeta_{j}-\zeta_{k}|:1\leq
j<k\leq\ell\right)$

then the open balls $B(\zeta_{1},\varepsilon)$, …,
$B(\zeta_{\ell},\varepsilon)$ are pairwise disjoint.

###### Proof.

It suffices to observe that if $z\in B(\zeta_{j},\varepsilon)\cap
B(\zeta_{k},\varepsilon)$, then

$2\varepsilon\leq|\zeta_{j}-\zeta_{k}|\leq|\zeta_{j}-z|+|z-\zeta_{k}|<2\varepsilon.$

which is absurd. This completes the proof. ∎

Let $\zeta_{1},\ldots,\zeta_{\ell}$ be the distinct roots of the polynomial
$f(z)$ and let $\mu_{j}$ be the multiplicity of the root $\zeta_{j}$ for all
$j\in\\{1,\ldots,\ell\\}$. Thus, $\sum_{j=1}^{\ell}\mu_{j}=n$ and

$f(z)=a_{n}\prod_{j=1}^{\ell}(z-\zeta_{j})^{\mu_{j}}$

with $a_{n}\neq 0$. Let $\varepsilon>0$. A polynomial $g(z)$ of degree $n$ is
_$\varepsilon$ -aligned_ to $f(z)$ if, for all $j\in\\{1,\ldots,\ell\\}$, the
open ball $B(\zeta_{j},\varepsilon)$ contains at least $\mu_{j}$ roots of
$g(z)$, counted with multiplicity.

If $\varepsilon$ satisfies inequality (1), then, by Lemma 1, the open balls
$B(\zeta_{1},\varepsilon)$, …, $B(\zeta_{\ell},\varepsilon)$ are pairwise
disjoint and a polynomial $g(z)$ of degree $n$ is $\varepsilon$-aligned to
$f(z)$ if and only if, for all $j\in\\{1,\ldots,\ell\\}$, the open ball
$B(\zeta_{j},\varepsilon)$ contains exactly $\mu_{j}$ roots of $g(z)$, counted
by multiplicity. Each root of $g(z)$ belongs to one and only one of the balls
$B(\zeta_{j},\varepsilon)$.

In Section 2 we prove that if $f(z)$ and $g(z)$ are polynomials of degree $n$
such that $g(z)$ is $\varepsilon$-aligned to $f(z)$ for all $\varepsilon>0$,
then $g(z)=cf(z)$ for some $c\neq 0$.

The theorem on the continuity of the roots of a polynomial states that for
every $\varepsilon>0$ there exists $\delta>0$ such that if the polynomial
$g(z)$ is a $\delta$-deformation of $f(z)$, then the polynomial $g(z)$ is
$\varepsilon$-aligned to $f(z)$. Section 3 considers the special case of
polynomials whose only root is zero and Section 4 considers polynomials that
have a nonzero root.

##  2\. The inverse theorem

We begin with a preliminary observation.

######  Theorem 1.

Let $f(z)$ and $g(z)$ be polynomials in $K[z]$ of degree $n$. Then $g(z)$ is
$\varepsilon$-aligned to $f(z)$ for all $\varepsilon>0$ if and only if
$g(z)=cf(z)$ for some $c\neq 0$.

###### Proof.

If $g(z)=cf(z)$ for some $c\neq 0$, then $f(z)$ and $g(z)$ have the same roots
with the same multiplicities and so $g(z)$ is $\varepsilon$-aligned to $f(z)$
for all $\varepsilon>0$.

Suppose that $g(z)$ is $\varepsilon$-aligned to $f(z)$ for all
$\varepsilon>0$. Let $\zeta_{1},\ldots,\zeta_{\ell}$ be the distinct roots of
the polynomial $f(z)$ and let $\mu_{j}$ be the multiplicity of $\zeta_{j}$ for
all $j\in\\{1,\ldots,\ell\\}$. Let $\varepsilon$ satisfy inequality (1). The
open balls $B(\zeta_{1},\varepsilon),\ldots,B(\zeta_{\ell},\varepsilon)$ are
pairwise disjoint. For $j\in\\{1,\ldots,\ell\\}$, let
$\omega_{1},\ldots,\omega_{\mu_{j}}$ be the $\mu_{j}$ not necessarily distinct
roots of $g(z)$ that are in the open ball $B(\zeta_{j},\varepsilon)$. If
$0<\varepsilon^{\prime}<\varepsilon$, then
$B(\zeta_{j},\varepsilon^{\prime})\subseteq B(\zeta_{j},\varepsilon)$. Because
the ball $B(\zeta_{j},\varepsilon^{\prime})$ also contains exactly $\mu_{j}$
roots of $g(z)$, counted by multiplicity, we must have

$\\{\omega_{1},\ldots,\omega_{\mu_{j}}\\}\subseteq
B(\zeta_{j},\varepsilon^{\prime})$

and so

$|\omega_{i}-\zeta_{j}|<\varepsilon^{\prime}$

for all $i\in\\{1,\ldots,\mu_{j}\\}$ and $0<\varepsilon^{\prime}<\varepsilon$.
It follows that $\omega_{i}=\zeta_{j}$ for all $i\in\\{1,\ldots,\mu_{j}\\}$
and so the polynomials $f(z)$ and $g(z)$ have the same roots with the same
multiplicities. Therefore, $g(z)=cf(z)$ for some $c\neq 0$. This completes the
proof. ∎

The following result is an inverse to the theorem on continuity of roots.

######  Theorem 2.

Let $\delta>0$. There exists $\varepsilon=\varepsilon(\delta)>0$ such that if
the polynomials $f(z)$ and $g(z)$ of degree $n$ are $\varepsilon$-aligned,
then $g(z)$ is a $\delta$-deformation of $cf(z)$ for some $c\neq 0$.

###### Proof.

This follows from Viéte’s formulae for the coefficients of a polynomial as the
elementary symmetric functions of the roots. For $k\in\\{1,2,\ldots,n\\}$, the
$k$th elementary symmetric function of $n$ variables is the polynomial

$\sigma_{k}(z_{1},\ldots,z_{n})=\sum_{1\leq i_{1}<\cdots<i_{k}\leq
n}z_{i_{1}}\cdots z_{i_{k}}.$

We define $\sigma_{0}(z_{1},\ldots,z_{n})=1$.

Let

$f(z)=\sum_{i=0}^{n}a_{i}z^{i}=a_{n}\prod_{j=1}^{n}(z-\zeta_{j})$

with $a_{n}\neq 0$. Viéte’s formulae are the identities

$\frac{a_{i}}{a_{n}}=(-1)^{n-i}\sigma_{n-i}(\zeta_{1},\ldots,\zeta_{n})$

for all $i\in\\{0,1,\ldots,n\\}$.

Let $\varepsilon>0$ and let $g(z)$ be a polynomial of degree $n$ that is
$\varepsilon$-aligned to $f(z)$. We enumerate the roots
$\omega_{1},\ldots,\omega_{n}$ of $g(z)$ so that

$|\omega_{j}-\zeta_{j}|<\varepsilon$

for all $j\in\\{1,2,\ldots,n\\}$. Setting

$t_{j}=\omega_{j}-\zeta_{j}$

we have

$|t_{j}|<\varepsilon$

for all $j\in\\{1,2,\ldots,n\\}$. There exists $b_{n}\neq 0$ such that

$g(z)=b_{n}\prod_{j=1}^{n}(z-\omega_{j})=b_{n}\prod_{j=1}^{n}(z-\zeta_{j}-t_{j})=\sum_{i=0}^{n}b_{i}z^{i}$

where

$\frac{b_{i}}{b_{n}}=(-1)^{n-i}\sigma_{n-i}(\omega_{1},\ldots,\omega_{n})=(-1)^{n-i}\sigma_{n-i}(\zeta_{1}+t_{1},\ldots,\zeta_{n}+t_{n})$

for all $i\in\\{0,1,\ldots,n-1\\}$.

The symmetric functions are polynomials, hence continuous, and so there exists
$\varepsilon=\varepsilon(\delta)>0$ such that if $|t_{j}|<\varepsilon$ for all
$j\in\\{1,\ldots,n\\}$, then

$\left|\sigma_{n-i}(\zeta_{1}+t_{1},\ldots,\zeta_{n}+t_{n})-\sigma_{n-i}(\zeta_{1},\ldots,\zeta_{n})\right|<\frac{\delta}{|b_{n}|}.$

Equivalently,

$\left|\frac{b_{i}}{b_{n}}-\frac{a_{i}}{a_{n}}\right|<\frac{\delta}{|b_{n}|}$

and

$\left|b_{i}-ca_{i}\right|<\delta$

for $c=b_{n}/a_{n}$ and all $i\in\\{0,1,\ldots,n\\}$. Thus, $g(z)$ is a
$\delta$-deformation of $cf(z)$. This completes the proof. ∎

##  3\. Polynomials with only the zero root

In this section we prove the continuity of roots of polynomials that have no
nonzero root.

######  Theorem 3.

Let $f(z)=a_{n}z^{n}$ with $a_{n}\neq 0$. For all $\varepsilon>0$ there exists
$\delta>0$ such that if the polynomial

$g(z)=\sum_{i=0}^{n}b_{i}z^{i}$

is a $\delta$-deformation of $f(z)$, then $|\omega|<\varepsilon$ for all roots
$\omega$ of $g(z)$, that is, $g(z)$ is $\varepsilon$-aligned to $f(z)$.

###### Proof.

Let $0<\varepsilon<1$ and let

(2) $0<\delta<\frac{\varepsilon^{n}|a_{n}|}{2n}.$

If $g(z)=\sum_{i=0}^{n}b_{i}z^{i}$ is a $\delta$-deformation of $f(z)$, then

$|b_{n}-a_{n}|<\delta$

and

$|b_{i}|<\delta\qquad\text{for all $i\in\\{0,1,\ldots,n-1\\}$.}$

The inequality

$|a_{n}|-|b_{n}|\leq|b_{n}-a_{n}|<\delta<\frac{|a_{n}|}{2n}$

implies

$|b_{n}|>|a_{n}|-\frac{|a_{n}|}{2n}\geq\frac{|a_{n}|}{2}>0.$

Therefore,

(3) $\frac{|b_{i}|}{|b_{n}|}<\frac{\delta}{|a_{n}|/2}=\frac{2\delta}{|a_{n}|}$

for all $i\in\\{0,1,\ldots,n-1\\}$.

If $|\omega|\geq 1$ and

$g(\omega)=\sum_{i=0}^{n}b_{i}\omega^{i}=0$

then

$\displaystyle 0<\frac{|a_{n}|}{2}|\omega|^{n}$
$\displaystyle<|b_{n}||\omega|^{n}=\left|-b_{n}\omega^{n}\right|=\left|\sum_{i=0}^{n-1}b_{i}\omega^{i}\right|$
$\displaystyle\leq\sum_{i=0}^{n-1}\left|b_{i}\right|\left|\omega\right|^{i}<\delta\sum_{i=0}^{n-1}\left|\omega\right|^{i}\leq
n\delta|\omega|^{n-1}$

and so

$|\omega|<\frac{2n\delta}{|a_{n}|}<\varepsilon^{n}<1$

which is absurd. Therefore, if $\omega\neq 0$ and $g(\omega)=0$, then
$0<|\omega|<1$.

If $0<|\omega|<1$ and
$g(\omega)=b_{n}\omega^{n}+\sum_{i=0}^{n-1}b_{i}\omega^{i}=0$, then

$-1=\sum_{i=0}^{n-1}\frac{b_{i}\omega^{i}}{b_{n}\omega^{n}}=\sum_{i=0}^{n-1}\frac{b_{i}}{b_{n}}\frac{1}{\omega^{n-i}}.$

Applying inequalities (2) and (3), we obtain

$\displaystyle 1$
$\displaystyle\leq\sum_{i=0}^{n-1}\frac{|b_{i}|}{|b_{n}|}\frac{1}{|\omega|^{n-i}}<\frac{2\delta}{|a_{n}|}\sum_{i=0}^{n-1}\frac{1}{|\omega|^{n-i}}$
$\displaystyle\leq\left(\frac{2n\delta}{|a_{n}|}\right)\frac{1}{|\omega|^{n}}<\left(\frac{\varepsilon}{|\omega|}\right)^{n}$

and so $|\omega|<\varepsilon$. This completes the proof. ∎

##  4\. Polynomials with nonzero roots

We begin with a simple result that implies continuity of roots of a polynomial
with only simple roots, that is, roots of multiplicity one.

######  Theorem 4.

Let $f(z)$ be a polynomial of degree $n$. For every $\varepsilon>0$, there
exists $\delta=\delta(f)>0$ such that if $g(z)$ is a polynomial of degree
$n\geq 1$ that is a $\delta$-deformation of $f(z)$, then for every root
$\zeta$ of $f(z)$ there is a root $\omega$ of $g(z)$ such that
$|\omega-\zeta|<\varepsilon$.

Note that $\delta(f)$ depends only on the polynomial $f(z)$ and not on $g(z)$.

###### Proof.

Let

$f(z)=\sum_{i=0}^{n}a_{i}z^{i}\in K[z]$

be a polynomial of degree $n$ with distinct roots
$\zeta_{1},\ldots,\zeta_{\ell}$ and let

$M=\max(1,|\zeta_{1}|,\ldots,|\zeta_{\ell}|).$

For $0<\varepsilon<M$, let

(4) $0<\delta<\frac{|a_{n}|}{2(n+1)}\left(\frac{\varepsilon}{M}\right)^{n}.$

Let

$g(z)=\sum_{i=0}^{n}b_{i}z^{i}\in K[z]$

be a polynomial of degree $n$ that is a $\delta$-deformation of $f(z)$. The
inequality

$|b_{n}-a_{n}|<\delta<\frac{|a_{n}|}{2}$

implies

$\frac{|a_{n}|}{2}<|b_{n}|.$

Let

$g(z)=b_{n}\prod_{j=1}^{n}(z-\omega_{j})$

where $\omega_{1},\ldots,\omega_{n}$ are the $n$ not necessarily distinct
roots of $g(z)$. For all $k\in\\{1,\ldots,\ell\\}$ we have $f(\zeta_{k})=0$
and so

$b_{n}\prod_{j=1}^{n}(\zeta_{k}-\omega_{j})=g(\zeta_{k})=g(\zeta_{k})-f(\zeta_{k})=\sum_{i=0}^{n}(b_{i}-a_{i})\zeta_{k}^{i}.$

Therefore,

$\displaystyle\frac{|a_{n}|}{2}\left(\min_{1\leq j\leq
n}|\zeta_{k}-\omega_{j}|\right)^{n}$
$\displaystyle\leq|b_{n}|\prod_{j=1}^{n}|\zeta_{k}-\omega_{j}|$
$\displaystyle\leq\sum_{i=0}^{n}|b_{i}-a_{i}||\zeta_{k}|^{i}$
$\displaystyle\leq(n+1)\delta M^{n}$

and so there exists $j_{k}\in\\{1,\ldots,n\\}$ such that

$|\zeta_{k}-\omega_{j_{k}}|^{n}\leq\frac{2(n+1)\delta M^{n}}{|a_{n}|}.$

Equivalently,

$|\zeta_{k}-\omega_{j_{k}}|\leq\left(\frac{2(n+1)\delta}{|a_{n}|}\right)^{1/n}M<\varepsilon.$

This completes the proof. ∎

######  Lemma 2.

Let

$f(z)=\sum_{i=0}^{n}a_{i}z^{i}\in K[z]$

be a polynomial of degree $n$ and let $\zeta$ be a nonzero root of $f(z)$.
Then

$\hat{f}(z)=\frac{f(z)}{z-\zeta}=\sum_{i=0}^{n-1}\hat{a}_{i}z^{i}$

is a polynomial in $K[z]$ of degree $n-1$ and

$\hat{a}_{i}=-\frac{1}{\zeta^{i+1}}\sum_{k=0}^{i}a_{k}\zeta^{k}$

for all $i\in\\{0,1,\ldots,n-1\\}$.

###### Proof.

This is a straightforward calculation. We have

$\displaystyle\sum_{i=0}^{n}a_{i}z^{i}$
$\displaystyle=f(z)=(z-\zeta)\hat{f}(z)=(z-\zeta)\sum_{i=0}^{n-1}\hat{a}_{i}z^{i}$
$\displaystyle=-\zeta\hat{a}_{0}+\sum_{i=1}^{n-1}\left(\hat{a}_{i-1}-\zeta\hat{a}_{i}\right)z^{i}+\hat{a}_{n-1}z^{n}$

and so

$a_{i}=\begin{cases}-\zeta\hat{a}_{0}&\text{if $i=0$}\\\
\hat{a}_{i-1}-\zeta\hat{a}_{i}&\text{if $i\in\\{1,\ldots,n-1\\}$}\\\
\hat{a}_{n-1}&\text{if $i=n$.}\end{cases}$

Equivalently,

$\hat{a}_{0}=-\frac{a_{0}}{\zeta}$

and

$\hat{a}_{i}=-\frac{1}{\zeta}\left(a_{i}-\hat{a}_{i-1}\right)\qquad\text{for
$i\in\\{1,\ldots,n-1\\}$.}$

We obtain

$\hat{a}_{1}=-\frac{1}{\zeta}(a_{1}-\hat{a}_{0})=-\frac{1}{\zeta^{2}}\left(a_{0}+a_{1}\zeta\right).$

If $i\in\\{1,2,\ldots,n-1\\}$ and

$\hat{a}_{i-1}=-\frac{1}{\zeta^{i}}\sum_{k=0}^{i-1}a_{k}\zeta^{k}$

then

$\displaystyle\hat{a}_{i}$
$\displaystyle=-\frac{1}{\zeta}\left(a_{i}-\hat{a}_{i-1}\right)=-\frac{1}{\zeta}\left(a_{i}+\frac{1}{\zeta^{i}}\sum_{k=0}^{i-1}a_{k}\zeta^{k}\right)$
$\displaystyle=-\frac{1}{\zeta^{i+1}}\sum_{k=0}^{i}a_{k}\zeta^{k}.$

This completes the proof. ∎

######  Lemma 3.

Let

$f(z)=\sum_{i=0}^{n}a_{i}z^{i}\in K[z]$

and

$g(z)=\sum_{i=0}^{n}b_{i}z^{i}\in K[z]$

be polynomials of degree $n$. Let $\zeta_{1}$ be a nonzero root of $f(z)$, let
$\omega_{1}$ be a nonzero root of $g(z)$, and let

$\hat{f}(z)=\frac{f(z)}{z-\zeta_{1}}=\sum_{i=0}^{n-1}\hat{a}_{i}z^{i}$

and

$\hat{g}(z)=\frac{g(z)}{z-\omega_{1}}=\sum_{i=0}^{n-1}\hat{b}_{i}z^{i}.$

For every $\lambda>0$ there exists $\kappa>0$ such that if $g(z)$ is a
$\kappa$-deformation of $f(z)$ and if

$|\omega_{1}-\zeta_{1}|<\kappa$

then $\hat{g}(z)$ is a $\lambda$-deformation of $\hat{f}(z)$.

###### Proof.

Let $K^{\times}=K\setminus\\{0\\}$. For all $i\in\\{0,1,\ldots,n-1\\}$, the
function

$\varphi_{i}:K^{i+1}\times K^{\times}\rightarrow K$

defined by

$\varphi_{i}(u_{0},u_{1},\ldots,u_{i},v)=-\frac{1}{v^{i+1}}\sum_{k=0}^{i}u_{k}v^{k}$

is continuous. Let

$(a_{0},a_{1},\ldots,a_{n},\zeta_{1})\in K^{n+1}\times K^{\times}.$

There exists $\kappa>0$ such that if

$(u_{0},u_{1},\ldots,u_{n},v)\in K^{n+1}\times K^{\times}$

and if

$|u_{i}-a_{i}|<\kappa$

for all $i\in\\{0,1,\ldots,n\\}$ and

$|v-\zeta_{1}|<\kappa$

then

$\left|\varphi_{i}(u_{0},u_{1},\ldots,u_{i},v)-\varphi_{i}(a_{0},a_{1},\ldots,a_{i},\zeta_{1})\right|<\lambda$

for all $i\in\\{0,1,\ldots,n-1\\}$.

By Lemma 2,

$\varphi_{i}(a_{0},a_{1},\ldots,a_{i},\zeta_{1})=\hat{a}_{i}$

and

$\varphi_{i}(b_{0},b_{1},\ldots,b_{i},\omega_{1})=\hat{b}_{i}$

for all $i\in\\{0,1,\ldots,n-1\\}$. If $g(z)$ is a $\kappa$-deformation of
$f(z)$, then

$|b_{i}-a_{i}|<\kappa$

for all $i\in\\{0,1,\ldots,n\\}$. The inequality

$|\omega_{1}-\zeta_{1}|<\kappa$

implies

$|\hat{b}_{i}-\hat{a}_{i}|=\left|\varphi_{i}(b_{0},b_{1},\ldots,b_{i},\omega_{1})-\varphi_{i}(a_{0},a_{1},\ldots,a_{i},\zeta_{1})\right|<\lambda$

for all $i\in\\{0,1,\ldots,n-1\\}$ and so $\hat{g}(z)$ is a
$\lambda$-deformation of $\hat{f}(z)$. This completes the proof. ∎

Here is the continuity of roots theorem for polynomials with a nonzero root.

######  Theorem 5.

Let $f(z)\in K[z]$ be a polynomial of degree $n\geq 1$ with a nonzero root.
For all $\varepsilon>0$, there exists $\delta=\delta(\varepsilon)>0$ such that
if $g(z)\in K[z]$ is a polynomial of degree $n$ that is a $\delta$-deformation
of $f(z)$, then $g(z)$ is $\varepsilon$-aligned to $f(z)$.

###### Proof.

Let

$f(z)=\sum_{i=0}^{n}a_{i}z^{i}\in K[z].$

Let $\zeta_{1},\ldots,\zeta_{\ell}$ be the distinct roots of $f(z)$ with
$\zeta_{1}\neq 0$. Without loss of generality we can assume that
$0<\varepsilon\leq|\zeta_{1}|$ and that $\varepsilon$ satisfies inequality
(1).

The proof is by induction on the degree $n$ of $f(z)$. The case $n=1$ follows
from Theorem 4. Let $n\geq 2$ and assume that Theorem 5 is true for
polynomials of degree $n-1$.

The polynomial

$\hat{f}(z)=\frac{f(z)}{z-\zeta_{1}}=\sum_{i=0}^{n-1}\hat{a}_{i}z^{i}$

has degree $n-1$. By the induction hypothesis, there exists
$\lambda=\lambda(\varepsilon)>0$ such that if $h(z)\in K[z]$ is a polynomial
of degree $n-1$ that is a $\lambda$-deformation of $\hat{f}(z)$, then $h(z)$
is $\varepsilon$-aligned to $\hat{f}(z)$. It suffices to prove that there
exists $\delta>0$ such that if the polynomial

$g(z)=\sum_{i=0}^{n}b_{i}z^{i}\in K[z]$

is a $\delta$-deformation of $f(z)$, then $g(z)$ has a nonzero root
$\omega_{1}$ with $|\omega_{1}-\zeta_{1}|<\varepsilon$ such that the
polynomial of degree $n-1$

$\hat{g}(z)=\frac{g(z)}{z-\omega_{1}}=\sum_{i=0}^{n-1}\hat{b}_{i}z^{i}$

is a $\lambda$-deformation of $\hat{f}(z)$.

Let $0<\kappa<\varepsilon$. By Theorem 4, there exists
$\delta_{1}=\delta_{1}(\kappa)>0$ such that if

$g(z)=\sum_{i=0}^{n}b_{i}z^{i}\in K[z]$

is a polynomial of degree $n$ that is a $\delta_{1}$-deformation of $f(z)$,
then there is a root $\omega_{1}$ of $g(z)$ such that

$|\omega_{1}-\zeta_{1}|<\kappa<\varepsilon.$

The inequality $\varepsilon\leq|\zeta_{1}|$ implies $\omega_{1}\neq 0$.

By Lemma 3, there exists $0<\kappa<\varepsilon$ such that if the polynomial
$g(z)$ is a $\kappa$-deformation of $f(z)$ and if $\omega_{1}$ is a nonzero
root of $g(z)$ with $|\omega_{1}-\zeta_{1}|<\kappa$, then $\hat{g}(z)$ is a
$\lambda$-deformation of $\hat{f}(z)$.

Let $\delta=\min(\kappa,\delta_{1})$. If $g(z)$ is a $\delta$-deformation of
$f(z)$, then $g(z)$ has a nonzero root $\omega_{1}$ with
$|\omega_{1}-\zeta_{1}|<\kappa$ such that $\hat{g}(z)$ is a
$\lambda$-deformation of $\hat{f}(z)$. It follows that $\hat{g}(z)$ is a
$\varepsilon$-aligned to $\hat{f}(z)$.

Let $\zeta_{1},\zeta_{2},\ldots,\zeta_{n}$ be the roots of $f(z)$ and let
$\omega_{1},\omega_{2},\ldots,\omega_{n}$ be the roots of $g(z)$. We have

$\hat{f}(z)=\frac{f(z)}{z-\zeta_{1}}=a_{n}\prod_{j=2}^{n}(z-\zeta_{j})$

and

$\hat{g}(z)=\frac{g(z)}{z-\omega_{1}}=b_{n}\prod_{j=2}^{n}(z-\omega_{j}).$

The polynomials $\hat{f}(z)$ and $\hat{g}(z)$ are $\varepsilon$-aligned.
Because

$|\omega_{1}-\zeta_{1}|<\kappa<\varepsilon$

it follows that the polynomals $f(z)$ and $g(z)$ are $\varepsilon$-aligned.
This completes the proof. ∎

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