A Probabilistic Version of Eneström–Kakeya Theorem for Certain Random Polynomials

Abstract

This paper presents a comprehensive exploration of a probabilistic adaptation of the Eneström–Kakeya theorem, applied to random polynomials featuring various coefficient distributions. Unlike the deterministic rendition of the theorem, our study dispenses with the necessity of any specific coefficient order. Instead, we consider coefficients drawn from a spectrum of sets with diverse probability distributions, encompassing finite, countable, and uncountable sets. Furthermore, we provide a result concerning the probability of failure of Schur stability for a random polynomial with coefficients distributed independently and identically as standard normal variates. We also provide simulations to corroborate our results.

1. Introduction

Ascertaining the bounds of polynomial roots constitutes an intriguing and crucial collection of problems in the domains of polynomial theory and complex analysis. The Fundamental Theorem of Algebra informs us that every polynomial $P:\mathbb{C}\to \mathbb{C}$,

$$P\left(z\right)=c_0+c_{1}z+\cdots +c_n{z}^n,$$

(1)

of degree n possesses precisely n roots. These roots, when viewed geometrically, correspond to n points in the Argand plane. In the numerical computation algorithms of these roots, an a priori indication of the region where these roots may lie holds paramount importance. When the coefficients are discernible, we have a range of classical and contemporary results that provide bounds for all or specific roots, such as real roots or roots whose modulus exceeds 1. The Eneström–Kakeya theorem, a renowned result in the theory of polynomials and complex analysis, offers bounds on the location of polynomial roots with non-negative coefficients. It was independently proved by Gustav Eneström [1] in 1893 and Kakeya [2] in 1912. The Eneström–Kakeya theorem has several important extensions and applications, such as in the study of the stability of discrete-time systems and in the construction of error-correcting codes [3,4]. The classical Eneström–Kakeya theorem is usually stated as follows:

Theorem 1 (Eneström–Kakeya Theorem).

If $P\left(z\right)={\sum }_{\nu =0}^n{c}_{\nu }{z}^{\nu }$ is a polynomial of degree n with real coefficients satisfying $0\le c_0\le c_1\le \cdots \le c_n$, then all the zeros of P lie in $\left|z\right|\le 1.$

In 2012, Aziz and Zargar [5] proved the following generalisation of Theorem 1.

Theorem 2.

Let $P\left(z\right)={\sum }_{\nu =0}^n{c}_{\nu }{z}^{\nu }$ be a polynomial of degree n. If for some positive numbers k and s with $k\ge 1$, $0<s\le 1$, $0\le s{c}_0\le c_1\le \cdots \le c_{n-1}\le k{c}_n.$ Then, all the zeros of $P\left(z\right)$ lie in the disc

$$|z+k-1|\le k+\frac{2c_0}{c_n}(1-s).$$

(2)

In 2015, E. R. Nwaeze [6] proved the following generalisation of Theorem 2.

Theorem 3.

Let $P\left(z\right)={\sum }_{\nu =0}^n{c}_{\nu }{z}^{\nu }$ be a polynomial of degree n. If for some real numbers α and $\beta ,$ $c_0-\beta \le c_1\le \cdots \le c_{n-1}\le c_n+\alpha ,$ then all the zeros of $P\left(z\right)$ lie in the disc

$$\left|z+\frac{\alpha }{c_n}\right|\le \frac{1}{|c_n|}\left(c_n+\alpha -c_0+\beta +\left|\beta \right|+|c_0|\right).$$

(3)

The study of the hole probability is a a significant problem in the theory of random analytic functions. In a recent paper, Kuryaliak and Skaskiv consider a random entire function of the form $f(z,\omega )={\sum }_{n=0}^{+\infty }{\epsilon }_n\left({\omega }_1\right)\times {\xi }_n\left({\omega }_2\right)f_n{z}^n$, where $\left({\epsilon }_n\right)$ is a sequence of independent Steinhaus random variables, $\left({\xi }_n\right)$ is a sequence of independent standard complex Gaussian random variables, and a sequence of numbers $f_n\in \mathbb{C}$ is such that ${\overline{lim}}_{n\to +\infty }\sqrt[n]{\left|f_n\right|}=0$ and $\#\left\{n:f_n\ne 0\right\}=+\infty$ and obtain asymptotic estimates for the probability of the absence of zeros [7]. While polynomials are indeed special analytic functions, the properties of their distribution and the number of zeros cannot always be inferred from each other. As an example, consider the following two functions:

$$\begin{array}{cc}\hfill h_n\left(z\right)& =1+z+\frac{z^2}{2!}+\cdots +\frac{z^n}{n!}\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathrm{Exp}\left(z\right)& =1+z+\frac{z^2}{2!}+\cdots +\hfill \end{array}$$

(5)

While $\mathrm{Exp}\left(z\right)$ is indeed the limit of the series in Equation (4) as n goes to infinity, $h_n\left(z\right)=0$ has exactly n zeros for a given n, but ${lim}_{n\to \infty }{h}_n\left(z\right)=\mathrm{Exp}\left(z\right)$ has no zeros in the complex plane. In this paper, we take an elementary yet elegant approach to investigate the lower bounds on the probability of a random polynomial having all roots in the unit disc. The problem of exactly finding this probability is a very formidable problem and this work is an initiatory attempt to obtain some partial results and insights.

The remainder of this paper is structured as follows: In Section 2, we introduce a probabilistic version of the Eneström–Kakeya theorem, where the coefficients of the polynomial are chosen randomly from different sets with different probability distributions. We also explore the probability of finding the roots of the random polynomial inside the unit disc and derive several interesting results for different sets and distributions. We consider the case where the coefficients are chosen from a finite set of positive integers and show that the probability of all roots lying inside the unit disc can be expressed in terms of the number of solutions to a certain equation. We also consider the case where the coefficients are chosen from a continuous distribution and derive an expression for the probability of the roots lying inside the unit disc. Section 3 investigates the concept of Schur stability, in view of its close relationship with the location of the polynomial roots within the unit disc. A result concerning the likelihood of a random polynomial, with independently and identically distributed ($i.i.d.$) coefficients following a normal distribution, not maintaining Schur stability is presented. Finally, Section 4 provides an exhaustive discussion of the findings and executes simulations to observe the behaviour of random polynomial roots in relation to their position within the unit disc.

2. Probabilistic Versions of Eneström–Kakeya Theorem

In this section, we study the probabilistic version of the Enström–Kakeya theorem concerning random polynomials for discrete and continuous distributions on the coefficients. Polynomials with integer coefficients, in addition to being essential in computer algebra systems for performing exact arithmetic, play a central role in Diophantine equations, which are polynomial equations that seek integer solutions. The case of continuous coefficients is dealt with in Theorem 5.

Theorem 4.

Let the polynomial

$${\displaystyle P\left(z\right)=\sum _{i=0}^n{c}_i\left(\omega \right)z^i}$$

(6)

be defined on the probability space $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$, where Ω is the sample space of sequences $\omega =({\omega }_0,{\omega }_1,\cdots ,{\omega }_n)$ with each ${\omega }_i\in S=\{1,2,\dots ,m\}$ for $m\in \mathbb{N}$; $\mathcal{F}$ is the sigma-algebra generated by the subsets of S; and $\mathbb{P}$ is the product measure arising from the i.i.d. uniform distribution over S. Then, the probability that the roots of the polynomial $P\left(z\right)$ lie in the disc $\mathbb{D}:\left|z\right|\le 1$ satisfies the inequality:

$$\mathbb{P}\left(\mathit{roots}\mathit{lying}\mathit{in}\mathbb{D}\right)\ge \frac{1}{m^{n+1}}\left(\genfrac{}{}{0pt}{}{m+n}{n}\right).$$

(7)

Proof. 

Consider the polynomial

$${\displaystyle P\left(z\right)=\sum _{i=0}^n{c}_i\left(\omega \right)z^i}$$

defined on the probability space $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$, where the coefficients $c_i\left(\omega \right)$ are independent and uniformly distributed over the set $S=\{1,2,\dots ,m\}$ for $m\in \mathbb{N}$.

For the condition $c_0\left(\omega \right)\ge c_1\left(\omega \right)\ge \cdots \ge c_n\left(\omega \right)$ to hold, we choose an n-element multi-set from S. The number of such multi-sets is the number of solutions to the equation

$$x_0+x_1+\cdots +x_n=m,x_i\ge 0,$$

(8)

which is given by

$$\left(\genfrac{}{}{0pt}{}{m+n}{n}\right).$$

The total number of ways to choose $(n+1)$ coefficients from S is $m^{n+1}$.

If the proposition “$A\Rightarrow B$” holds true, and if A occurs with probability p, then B occurs with a probability of at least p. Therefore, the probability that all the roots of the random polynomial $P\left(z\right)$ lie within the disc $\mathbb{D}:\left|z\right|\le 1$ is bounded from below as

$$\mathbb{P}\left(\mathrm{roots}\mathrm{lying}\mathrm{in}\mathbb{D}\right)\ge \frac{1}{m^{n+1}}\left(\genfrac{}{}{0pt}{}{m+n}{n}\right).$$

This completes the proof. □

Theorem 5.

Let the the polynomial

$${\displaystyle P\left(z\right)=\sum _{i=0}^n{ϵ}_i\left(\omega \right)z^i}$$

(9)

be defined on the probability space $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$, where Ω is the sample space of sequences $\omega =({\omega }_0,{\omega }_1,\cdots ,{\omega }_n)$ with ${\omega }_i\in [a,b],a>0,b>0$ for all i; $\mathcal{F}$ is the sigma-algebra generated by the Borel subsets of [a, b]; and $\mathbb{P}$ is the product measure arising from the i.i.d. densities f. The coefficients ${\left\{{ϵ}_i\right\}}_{i=0}^n$ are independent and identically distributed nonnegative random variables with probability density function f. Then, the probability that the roots of the polynomial lie inside the unit disc is at least ${\displaystyle \frac{1}{(n+1)!}}$.

Proof. 

Define the random variables ${ϵ}_i\left(\omega \right):\mathsf{\Omega }\to [a,b]$ by

$${ϵ}_i\left(\omega \right)={\omega }_i.$$

(10)

These are the coefficients of the polynomial.

Consider the event

$$A=\{{ϵ}_0\left(\omega \right)\ge {ϵ}_1\left(\omega \right)\ge \cdots \ge {ϵ}_n\left(\omega \right)\}.$$

(11)

Given that the coefficients ${\left\{{ϵ}_i\right\}}_{i=0}^n$ are independently and identically distributed (i.i.d.), we can reason as follows:

Any specific ordering of the sequence $({ϵ}_0,{ϵ}_1,\cdots ,{ϵ}_n)$ has probability $\frac{1}{(n+1)!}$. This is because there are $(n+1)!$ possible ways to order these $n+1$ terms, and each of them is equally likely due to the i.i.d. nature of the sequence.

Considering the event A,

$$\mathbb{P}\left(A\right)=\mathbb{P}({ϵ}_0\ge {ϵ}_1\ge \cdots \ge {ϵ}_n)$$

(12)

Under the given assumptions, $\mathbb{P}\left(A\right)$ can be written as

$$\mathbb{P}\left(A\right)={\int }_{\mathsf{\Omega }}{1}_A\left(\omega \right)d\mathbb{P}\left(\omega \right)$$

(13)

where $1_A\left(\omega \right)$ is the indicator function for the event A. Now, to evaluate the integral in Equation (13), we can reason as follows:

Given the total number of possible orderings for the i.i.d. random variables ${\left\{{ϵ}_i\right\}}_{i=0}^n$ is $(n+1)!$, and assuming the probability distribution $f\left(x\right)$ is symmetric about its mean (because the variables are exchangeable), the probability p for any one of these orderings is

$$p=\frac{1}{(n+1)!}.$$

(14)

Thus, for a random polynomial of degree n with coefficients from this distribution, the probability of the zeros lying inside the unit disc is at least ${\displaystyle \frac{1}{(n+1)!}}$.

This completes the proof. □

Remark 1.

The above lower bound holds irrespective of the distribution on the coefficients as long as the coefficients are independently and identically distributed. Unfortunately, the bound on the probability does not give us any useful information about the location of roots in the unit disc for large-degree polynomials. However, in Theorem 6, we use an entirely different approach based on the probabilistic rules of inference, which exploits Vieta’s formula for the product of roots of a polynomial in terms of the coefficients.

Remark 2.

It is also pertinent to mention here that the converse of Eneström–Kekaya does not hold in general. In other words, polynomials $P\left(z\right)={\sum }_{\nu =0}^n{c}_{\nu }{z}^{\nu }$ satisfying the condition $0\le c_0\le c_1\le \cdots \le c_n$ are not the only polynomials that have zeros inside the unit disc $\left|z\right|<1.$ As an illustration, consider the polynomial with roots ${\displaystyle \frac{1}{2}}$ and ${\displaystyle \frac{1}{3}}$

$$w\left(z\right)=z^2-{\displaystyle \frac{5}{6}}z+{\displaystyle \frac{1}{6}}$$

3. Schur Stability of Random Polynomials

The stability of a polynomial is a concept related to the location of its roots in the complex plane. While there are different definitions of stability depending on the context, the two main definitions often employed are Schur stability and Hurwitz stability, which are related to discrete-time and continuous-time systems, respectively [4]. However, in this article, we will confine ourselves to Schur stability as it has a bearing on our results. A polynomial is Schur stable if all its roots lie strictly inside the unit circle in the complex plane. Formally, we have the following definition:

Definition 1.

Consider a polynomial $P\left(z\right)$ of degree n:

$$P\left(z\right)=c_0+c_{1}z+c_2{z}^2+\cdots +c_n{z}^n,$$

then $P\left(z\right)$ is Schur stable if all its roots $z_i$ satisfy $|z_i| <1.$

We prove the following theorem concerning the probability of failure of the Schur stability of random polynomials $P\left(z\right)={\sum }_0^n{c}_i{z}^i$ with $i.i.d$ and normally distributed coefficients.

Theorem 6.

Let the polynomial

$${\displaystyle P\left(z\right)=\sum _{i=0}^n{c}_i\left(\omega \right)z^i}$$

(15)

be defined on the probability space $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$, where Ω is the sample space of sequences $\omega =({\omega }_0,{\omega }_1,\cdots ,{\omega }_n)$ with each ${\omega }_i\in \mathbb{R}$; $\mathcal{F}$ is the sigma-algebra generated by the Borel subsets of $\mathbb{R}$; and $\mathbb{P}$ is the product measure arising from the $i.i.d.$ densities f. The coefficients ${\left\{c_i\left(\omega \right)\right\}}_{i=0}^n$ are independent and identically distributed random variables with a standard normal distribution. Then, the probability that $P\left(z\right)$ fails to be Schur stable is ${\displaystyle \frac{1}{2}}$.

Proof. 

Consider the polynomial

$${\displaystyle P\left(z\right)=\sum _{i=0}^n{c}_i\left(\omega \right)z^i}$$

(16)

defined on the probability space $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$, where the coefficients ${\left\{c_i\left(\omega \right)\right\}}_{i=0}^n$ are independent and identically distributed standard normal random variables.

We want to find the lower bounds on the probability that at least one root of the polynomial lies outside the disc $\left|z\right|\le 1$. Observe that if all roots of $P\left(z\right)$ are within the disc $\left|z\right|\le 1$, then the absolute value of the product of the roots should be less than or equal to 1. By Vieta’s formulae, the product of the roots is equal to ${(-1)}^{n-1}{\displaystyle \frac{c_0\left(\omega \right)}{c_n\left(\omega \right)}}$. Therefore, we have

$$\left|{(-1)}^{n-1}\right|\left|\frac{c_0\left(\omega \right)}{c_n\left(\omega \right)}\right|\le 1.$$

The event that all roots are within the disc $\left|z\right|\le 1$ is guaranteed by the event $\left|\frac{c_0\left(\omega \right)}{c_n\left(\omega \right)}\right|\le 1$. Because $c_0\left(\omega \right)$ and $c_n\left(\omega \right)$ are i.i.d. standard normal variates, their ratio follows a well-known Cauchy distribution [8,9]:

$${\displaystyle \frac{c_0\left(\omega \right)}{c_n\left(\omega \right)}}\sim \mathcal{C}(0,1).$$

The probability density function of the Cauchy distribution is given by

$$f\left(x\right)=\frac{1}{\pi (1+x^2)}.$$

To find a lower probability that all roots are within the disc $\left|z\right|\le 1$, we integrate the pdf over the interval $[-1,1]$ as

$$P\left(\left|\frac{c_0\left(\omega \right)}{c_n\left(\omega \right)}\right|\le 1\right)={\int }_{-1}^1\frac{1}{\pi (1+x^2)}dx.$$

Therefore, the probability that at least one root lies outside the disc is greater or equal to $P\left(\left|\frac{c_0\left(\omega \right)}{c_n\left(\omega \right)}\right|>1\right)$, which can be expressed as

$$P\left(\left|\frac{c_0\left(\omega \right)}{c_n\left(\omega \right)}\right|>1\right)=1-P\left(\left|\frac{c_0\left(\omega \right)}{c_n\left(\omega \right)}\right|\le 1\right)=1-{\int }_{-1}^1\frac{1}{\pi (1+x^2)}dx.$$

(17)

The integral in Equation (17) simplifies to

$${\int }_{-1}^1\frac{1}{\pi (1+x^2)}dx={\left[\frac{1}{\pi }arctan\left(x\right)\right]}_{-1}^1=\frac{1}{\pi }(arctan\left(1\right)-arctan(-1))=\frac{1}{\pi }(\frac{\pi }{4}-\frac{-\pi }{4})=\frac{1}{2}.$$

Thus, the probability that at least one root lies outside the disc $\left|z\right|\le 1$ is at least $\frac{1}{2}$. □

4. Discussion and Simulation

Theorem 5 establishes a lower bound on the probability that the roots of a random polynomial with i.i.d. coefficients lie inside the unit disc, which is ${\displaystyle \frac{1}{(n+1)!}}$. As one can note, this bound becomes very small as n becomes large, so it does not give useful information for large-degree polynomials. To provide a better understanding of the behaviour of roots for large-degree polynomials, we can consider examining the expected distribution of the roots. One such distribution that has been extensively studied is the circular law, which states that for a random polynomial with i.i.d. coefficients, the roots tend to be uniformly distributed over the unit disc in the complex plane as the degree of the polynomial tends to infinity [10,11,12,13]. It is also pertinent to mention that the probability of at least one root lying outside the unit disc does not depend on the degree of the polynomial. This might seem apparently contradictory to the circular law, which states that, with the increase in the degree of the polynomial, the distribution of the roots converges weakly to the uniform distribution on the unit circle. However, the circular law is not concerned with the probability of having roots outside the unit circle but rather with the distribution of roots as the degree of the polynomial increases. It is possible that the probability of having at least one root outside the unit disc increases with the degree of the random polynomial, but the circular law is still valid because it pertains to the distribution of the roots as the degree of the polynomial grows. In fact, that is what the simulations in Figure 1 indeed reveal. The circular law is more about the density and distribution of roots, while our result concerns a specific probability related to the location of the roots of random polynomials.

As depicted in Figure 2, a representation of a random polynomial of degree 1000 provides an illustrative example of our primary focus. Notably, the circular law implies that the roots of the random polynomial become more concentrated within the unit disc as the polynomial’s degree increases. Concurrently, the probability of a root existing outside the unit disc gradually diminishes and becomes infinitesimally small.

Furthermore, Figure 3 offers a visualisation of the variation in probability for a random polynomial with independent and identically distributed (i.i.d.) coefficients following a normal distribution to have all the roots within the unit disc, denoted as $\mathbb{D}$. According to the simulation, this probability exhibits a swift decline as the degree of the polynomial escalates.

Figure 4 displays the anticipated fraction of roots in the unit disc for a random polynomial with standard normal, exponential, and uniform distribution. These insights, depicted graphically, provide a nuanced understanding of the interaction between the degree of the polynomial and the location of its roots within the unit disc.

Lastly, Figure 5 depicts the domain colouring of the unit disc under the polynomial map $f\left(z\right)=1+2z+5z^2+9z^3+11z^4+13z^5+16z^6.$ Obviously, $f\left(z\right)$ satisfies the hypothesis of the Enström–Kakeya theorem, and as such, all zeros lie inside $\left|z\right|<1.$ In the figure, all the six zeros are reflected by six dark spots in the domain colouring of the unit disc under the map. Domain colouring is a visualisation technique used in mathematics to represent complex functions using colour, where the hue often represents the argument (phase or angle) and the brightness or saturation represents the magnitude (absolute value) of the function’s output.

Discussion on the Role of Probability Distributions

The nature of probability distributions plays a pivotal role in the behaviour of the roots of random polynomials. Randomness introduces a new dimension of complexity to the realm of polynomial equations. With deterministic coefficients, the location of roots is fixed and can be evaluated deterministically. In the world of random polynomials, however, the roots are random variables themselves, and their distribution and behaviour depend heavily on the underlying coefficient distributions.

Consider the case where the coefficients are drawn from a normal distribution, which is symmetric around its mean. The behaviour is significantly different than when coefficients are derived from, say, an exponential distribution which is skewed. Based on our extensive simulations we find that the distribution of roots in the complex plane is influenced by the characteristics of the coefficient distributions:

1.

Mean and Variance: If the mean is not centred at zero or if the variance is high, this might induce higher chances of roots lying outside the unit disc, even for large-degree polynomials.

2.

Skewness: A skewed distribution might lead to roots clustering towards a certain part of the complex plane.

3.

Kurtosis: Distributions with high kurtosis might lead to more roots near the centre of the unit disc, with occasional roots far outside.

Our results, especially when we examine the behaviour for large-degree polynomials, demonstrate that the probabilistic nature of coefficients (as determined by their distributions) strongly dictates the behaviour of polynomial roots. When we say that roots tend to be uniformly distributed over the unit disc as the polynomial degree tends to infinity, this conclusion heavily relies on the assumption that the coefficients are independent and identically distributed. Changing the underlying coefficient distribution will invariably change this behaviour.

5. Potential Practical Applications

The insights gained from our exploration of random polynomial behaviour, while deeply rooted in theoretical mathematics, have several practical implications that extend beyond the realm of the theory of pure polynomials.

1.

Signal Processing: In the field of digital signal processing, the location of the roots of polynomials (often referred to as the zeros of the transfer function) can determine the stability of filters. Understanding the probabilistic behaviour of these roots can aid in designing robust filters under uncertain conditions.

2.

Control Systems: For dynamic systems, the location of polynomial roots can indicate the stability of the system. A probabilistic approach offers insights into the likelihood of system stability under variable conditions, especially in systems with inherent randomness or noise.

3.

Economics and Forecasting: In econometrics, polynomials are often used to model economic behaviours or phenomena. Knowing the probabilistic behaviour of polynomial roots can be instrumental in predicting uncertain economic futures, especially in volatile markets.

4.

Complex Systems Analysis: Many real-world systems, from neural networks to ecological systems, can be modelled as complex systems. A probabilistic understanding of polynomial behaviour can shed light on the emergent properties and behaviours of these systems under varying conditions.

5.

Cryptography: Random polynomials play a role in certain cryptographic systems. Understanding the distribution and behaviour of their roots can influence the design of secure communication protocols.

In essence, while our study begins as a theoretical exploration, its implications ripple through various domains, emphasising the profound impact of mathematics in shaping our understanding of complex, real-world systems.

6. Conclusions and Future Work

The research findings presented in this paper introduce a bridge between deterministic polynomial behaviour and probabilistic polynomial behaviour, emphasising the intricate dynamics introduced by randomness. While we have made some preliminary and initiatory attempts in understanding the behaviour of random polynomial roots and their relationship with respect to the unit disc, a multitude of avenues remain to be explored. However, a limitation of our method is that we have been only able to find the lower bounds on the probability that the roots of a random lie in the unit disc. The problem of exactly computing this probability for a given distribution in terms of the degree of the polynomial is a formidable problem. It is hoped that our work might give some researchers a head start on the problem and improve our bounds.

Future work could explore:

1.

Alternative Distributions: How do other distributions, like the Cauchy or Beta distribution, impact the behaviour of roots?

2.

Dependent Coefficients: How does the introduction of correlation or dependency among coefficients influence root distribution?

3.

Higher Dimensions: Extending the study to multivariate polynomials and exploring how randomness in coefficients affects the location of roots in multidimensional spaces.

4.

Generalised Theorems: Formulating and proving generalised versions of existing theorems in polynomial equations when randomness is introduced.

By examining the interplay between deterministic mathematical theorems and stochastic behaviour, we pave the way for an enriched understanding of polynomial equations and their solutions in the realm of polynomial theory. The work presented here serves as a stepping stone, and the domain promises myriad opportunities for further exploration.