Review Reports - More Symmetrical Solutions of Quadratic Equations: The Accompanying Variables Method

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I have read the manuscript in detail and conclude that this manuscript can be accepted for publication but needs major revision

1) The authors state that the method of accompanying variables is developed for quadratic polynomials that have no zeros and that zeros of associated vector polynomials are determined. The authors are requested to clarify why this particular method is necessary or advantageous for such polynomials, and to explain the underlying rationale for replacing the original scalar problem with an associated vector formulation.

2) Author state that "We now assume that there is a real variable y accompanying the real variable x, possibly hidden" on what basis did they assume if possoble kindly give proper references.

3) On page 9, the author wrote "as in (6.1)?" Kindly correct it.

4) The author must revise the reference section to be up-to-date. I see no reference to 2025.
5) The Abstract section should contain answers to the following questions: What problem was studied and why is it important? What are the important results? What conclusions can be drawn from the results? What is the novelty of the work and where does it go beyond previous efforts in the literature? The abstract section should be revised.

6) Strengthen the introduction by adding a concise review of recent publications related to the considered equation. This will help emphasize the novelty and current relevance of the study.

7) What are the physical applications of the results? This needs to be explained more.

8) References should continue in numerical order. Do not skip from [1] to [4].

9) The introduction section needs to be developed further. What does this approach solve according to the current complex number theory? In which applications is this method mandatory or advantageous?

10) Minor spelling errors and grammatical inconsistencies should be corrected.

The paper quality is very low and the overall presentation of the manuscript is significantly lacking. It requires major revision before it can be accepted.

The abstract is poorly written and does not clearly highlight the novelty of the study or adequately summarize the key results obtained. The author is advised to revise the abstract to explicitly emphasize the main contributions and findings of the paper.
There are several punctuation mistakes throughout the manuscript, like on page 2, the author writes “We consider the real-valued function with real coefficients. But they didn’t mention whether either sentence ends with a full stop or a comma. The author should revise and polish these errors consistently.
Throughout the manuscript, the author used too many writing styles author is advised to maintain uniformity throughout the manuscript.
Throughout the manuscript, many of the equations are not numbered and mostly are numbered on left hand side. The author is advised to assign equation numbers to all equations for clarity and ease of reference in right hand side
Page 9 Equation 5.5.“hO(z) = ?.” author are advised to must re-write this equation in standard format.
Throughout the entire paper, there are many grammatical errors that should be minutely corrected.
Theorem 5.1: The statement of the theorem must be revised.
Equation 4.7 has a full stop, but Equation 4.8 has no full stop, or the author must revise the manuscript in a very professional way.
Most sentences in the manuscript begin with “we.” The author is encouraged to revise the text to reduce the use of first-person pronouns.
The Comment part of the section is not well-written author is advised to rewrite this section in a professional way.

Author Response

Reply to Reviewer Report 1

In this paper, author developed a new method for analyzing quadratic polyno
mials without zeros by introducing accompanying variables and determining the
zeros of associated vector polynomials. They proposed a flexible vector-valued
squaring framework that can be adapted to various applications.
I have read the manuscript in detail and conclude that this manuscript can
be accepted for publication but needs major revision

1) The authors state that the method of accompanying variables is developed
for quadratic polynomials that have no zeros and that zeros of associated vector
polynomials are determined. The authors are requested to clarify why this par
ticular method is necessary or advantageous for such polynomials, and to explain
the underlying rationale for replacing the original scalar problem with an associ
ated vector formulation.

In order to provide the requested clarification, a clause was added to the first
sentence of the third section of the Introduction.
It follows directly from this that the classical solution method for quadratic
equations requires a replacement on a real basis.

2) Author state that ”We now assume that there is a real variable y accom
panying the real variable x, possibly hidden” on what basis did they assume if
possoble kindly give proper references.

In that sentence of the introduction which follows the one just discussed, the
close relationship between complex numbers and accompanying variables is high
lighted. Instead of introducing an imaginary quantity, which does not meet the
standards of highest mathematical rigor, it is therefore assumed that there is a
real quantity replacing it, which is connected to the original quantity by a vector
of R2. This is explained in the last sentense of Section 1 and explains where
Section 2 stops and Section 3 starts.

3) On page 9, the author wrote ”as in (6.1)?” Kindly correct it.

Did (see page 11, as in (37), corrected).

4) The author must revise the reference section to be up-to-date. I see no
reference to 2025.

The bibliography has been reordered. In Section 7, I discuss the only relevant
work from the year 2025 that I am aware of this topic. However, the recurring
appearance of an imaginary unit there is difficult to reconcile with the title of this
work. Therefore, I would appreciate specific suggestions for related 2025-papers

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The study did not yield any substantial novel insights. The study specifically examines the complex plane, or, correspondingly, the complex numbers, without presenting novel theoretical insights.

The document talks about the ring \[ \mathbb{C} = \bigl(\mathbb{R}^2,\;\oplus,\;\circledast,\;\cdot,\;\circ,\;1,\;I\bigr) \]

and asserts that it is isomorphic to the complex numbers in the context of rings.
Nonetheless, this assertion is articulated without adequate justification or innovative viewpoint.

The author also says on page 10, lines 3–4: “$\mathbb{C}$ is not an extension field of the real numbers because $\mathbb{R}$ is isomorphic to a subspace of $\mathbb{R}^2$ but not a subset.”
This claim is not true in algebra. The assertion that $\mathbb{R}$ is isomorphic to a subspace of $\mathbb{R}^2$ does not negate the possibility of $\mathbb{C}$ being a field extension of $\mathbb{R}$; this argument demonstrates a fundamental ignorance of field extensions and algebraic structures.

In conclusion, I urge that the work not be published in the journal \emph{Symmetry} because it is not innovative and contains severe conceptual and algebraic mistakes.

Author Response

$\bullet$
The study did not yield any substantial novel insights. The study specifically examines the complex plane, or, correspondingly, the complex numbers, without presenting novel theoretical insights.
\\

\textbf{Reply}

-The study does \textbf{not} examine the complex plane or complex numbers but examines a class of algebraic structures based upon the two-dimensional vector space $R^2$ and presents basic novel theoretical insight.

-In particular, this study avoids the use of any imaginary number. Many mathematical results involving imaginary numbers like i cannot be numerically evaluated because it is completely unknown what i is.

-This work describes solutions of quadratic equations which may belong to very general types of circles while classical solutions are restricted to Euclidean ones.

-In its simplest form, indeed, the present novel algebraic structure has a closer relationship, but by no means agreement, to the complex plane than is the case in the abstract generality.

-The complex plane makes use of an undefined, non existent, inexplicable, or just imaginary quantity i, whereas the present work makes use of the well-defined and easily verifyable two-dimensional real vector space $R^2$.
\\

\noindent$\bullet$ The document talks about the ring \[ {\mathbb{C}} = \bigl(R^2,\;\oplus,\;\ast,\;\cdot,\;\circ,\textbf{1},\textbf{I}\bigr) \]
and asserts that it is isomorphic to the complex numbers in the context of rings.
\\

\textbf{Reply}

- It is \textbf{not} said in this manuscript that ${\mathbb{C}}$ is isomorphic to the complex numbers.

- For defining an isomorphy $\varphi:R^2\rightarrow \Theta $ between the well defined set $R^2$ and the set $\Theta$ of complex numbers one needs to be able to give an exact definition of $\Theta.$ This, however, is impossible because of the completely unknown nature of the imaginary unit i. People just say that i is not a real number and satisfies an equation of undefined nature: $i^2=-1$. But: what is squaring the "object/symbol/formal quantity" i of unknown nature? \\

$\bullet$ Nonetheless, this assertion is articulated without adequate justification or innovative viewpoint.
\\

\textbf{Reply}

No again, this assertion is \textbf{not} articulated in the manuscript.
\\

$\bullet$The author also says on page 10, lines 3–4: “${\mathbb{C}}$ is not an extension field of the real numbers because $\mathbb{R}$ is isomorphic to a subspace of ${\mathbb{R}}^2$ but not a subset.”
This claim is not true in algebra.
\\

\textbf{Reply}

Certain statements in the literature may indeed be incorrect, but it is obvious:

${R}$ is not a subfield of ${R}^2$ because it is not a subset but a subspace of ${R}^2$. Therefore, \textit{by definition}, $R^2$ cannot be an extension field of $R$.

This can be verified using any mathematical dictionary. \\

$\bullet$ this argument demonstrates a fundamental ignorance of field extensions and algebraic structures.\\

\textbf{Reply}

I think I was able to refute that in the above reply.\\

$\bullet$The assertion that $\mathbb{R}$ is isomorphic to a subspace of ${R}^2$ does \textit{not} negate the possibility of $\mathbb{C}$ being a field extension of $\mathbb{R}$;
\\

\textbf{Reply}

\textit{Oh yes}, I think I was able to refute that in the above reply. The definition of extension field that I use is based on several mathematical dictionaries. I couldn't find any other definition anywhere.\\

$\bullet $In conclusion, I urge that the work not be published in the journal \emph{Symmetry} because it is not innovative and contains severe conceptual and algebraic mistakes.
\\

\textbf{Reply}

I believe I have been able to refute all the opposing arguments and hope for publication in Symmetry.

It is a great achievement of the journal Symmetry to enable the reader to become aware of the debate conducted here.\\

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Kindly find the comments in the attachment.

Comments for author File: Comments.pdf

Author Response

Reply to Reviewer Report

This paper introduces the accompanying variables method as a novel, geo
metrically grounded approach to solving real quadratic equations that lack real
roots. Rather than invoking complex numbers, the author constructs vector
valued analogues of quadratic polynomials in R2, using alternative multiplication
operations: circular (Euclidean), generalized circular (based on positively homo
geneous functionals, or phs-functionals), and hyperbolic. The resulting solutions
are symmetric vector pairs lying on circles, generalized circles, or hyperbola
soffering a fully real-analytic reinterpretation of classical complex roots. The
manuscript is well-structured, mathematically sound, and offers a fresh perspec
tive on a classical problem. With modest enhancements in motivation, notation
clarity, and contextual framing, it will be a valuable addition to the literature.
3. Suggestions for Improvement

While the framework is elegant, the paper does not demonstrate a practical
scenario where this method offers advantages over standard complex analysis. A
short example, e.g., modeling generalized harmonic signals in biomedical engi
neering using lp-circles, would strengthen the impact.
Reply
This purpose is served by the new Section 5.
2. Symbols such as 11 = (1, 0)T , II = (0, 1)T , , , and h may confuse
readers unfamiliar with the authors prior work. A notation table would improve
accessibility.
Reply
Such a list is now inserted after Definition 1.
3. The paper cites the authors previous work on lp-complex and hyperbolic
numbers but does not engage with broader literature (e.g., Clifford algebras,
bicomplex numbers). A brief comparative discussion would better situate the
contribution.
Reply
The idea of replacing further imaginary units (by further real vari
ables), such as those that appear in Clifford algebras and bicomplex numbers,
to solve higher-degree equations would, in my opinion, exceed the scope of this
work, may dilute the drama of replacing the famous i with an apparent game
involving several imaginary units for some readers, and may also be outside the
interest of a large group of readers (possibly including students studying to be
come a teacher) who might mainly be interested in quadratic equations. For this
reason, I would prefer not to mention Clifford algebras and bicomplex numbers
and hope for understanding.
4. Theorem 5.1 shows that under condition (2.1), no hyperbolic solutions exist
a stark contrast to the circular case. This difference deserves deeper interpreta
tion: Is it inherent to the hyperbolic product, or does it reflect a fundamental
asymmetry?
Reply
In formal analytical terms, it is a consequence of the difference be
tween the formulas (3.2) [now (6)] and (5.3) [now (32)] of circular and hyperbolic
vector-squaring, respectively. While x2−y2 can become negative, this is excluded
for x2 + y2. From a geometric perspective, this phenomenon is accompanied by
the fact that vector quadratures are restricted to quadrants, in the hyperbolic
case, but not in the circular case.
5. Correct minor typographical and formatting inconsistencies.
Reply
Done.
Dear Reviewer, thank you very much for all your valuable advice!

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

Please see the attached file.

Comments for author File: Comments.pdf

Author Response

• Introduction is lack of information about recent references related to the topics of this study.

Reply: I added the following sentense to the Introduction:

Known solutions are made much more precise

... and supplemented by new ... solutions.

• The author should add the main motivation of this study in the introduction section.

Reply:

• Theauthor should add the mathematical differences and novelties of this study compared to Hamilton’s
bireal numbers.

Reply: Added to the Introduction and Comments Sections.

• Please add comma or dot at the end of all equations. For example equations 2,3,4 and 7

Reply: Several commas added.

• The proof of Theorem 3.1 is not clear enough for readers. Please add some clarifications steps.

Reply: A new first line is introduced.

• The author should add the advantages of the current vector-space-based replacement of the imaginary

unit provide.

Reply: The following advantages are mentioned both in the Abstract and in the text:

The very general method of real (instead of imaginary) accompanying variables is developed.
The flexible vector method uses a wide variety of vector-valued squaring methods.
This method can be adopted to a wide range of application situations.
Known complex solutions are made much more precise by replacing the imaginary component.
Real variables can be numerically evaluated, what is not the case for imaginary quantities.

• If possible add one example to highlight the advantages of this study.

Reply: Please see new Section 5.

• The author should add the conclusion remark and expand it by suggesting future work

Reply: Please see new Section 8.

Thank you for your comments.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I have carefully read the revised manuscript, but I still have several concerns regarding both the manuscript and, in particular, the response letter. My concerns and suggestions are given below.

1) Author reply most to the comments just "Did" for example, comment 8,9 etc what author did, he didn't explain

2) In the previous report, my comment was "The Comment part of the section is not well-written, author is advised to rewrite this section in a professional way," and the author's reply is "A specific indication of what' not well-written' means here would be helpful". In my opinion comment section must be strong and supporting the study with results.

3) In the previous report my comment was "Equation 4.7 has a full stop, but Equation 4.8 has no full stop, or the
author must revise the manuscript in a very professional way." author's reply is "Equation (4.7) has no full stop, but Equation (4.8) has full stop. The meaning
of this comment is unclear." I was referring to an inconsistency in punctuation. Some displayed equations in the manuscript end with a full stop (.), while others do not (for example, Equations (4.7) and (4.8)). Authors are advised to maintain uniformity throughout the manuscript.

4) In previous report my comment was "Theorem 5.1: The statement of the theorem must be revised."author's reply "Identifying a specific deficiency would be very helpful here." In my opinion statement must be clear and support the proof in theorm 5.1 i saw a lack of clarity

5) in previous report, my comment was "Throughout the entire paper, there are many grammatical errors that
should be minutely corrected." author's reply "Providing, for example, five or more examples would be helpful." In my opinion while writing a manuscript authors should check the whole paper and try the best to error free the manuscript. For this manuscript, Let me present a few example to him in which the grammar or punctuation is not correct.

Author sentence "For a long time already, one
might have therefore the impression that the theory of the fundamental theorem
of algebra can be considered to be largely rounded off" correct sentence "For a long time already, one might therefore have the impression that the theory of the fundamental theorem of algebra can be considered largely rounded off"

6) In my previous report my comment was "The author must revise the reference section to be up-to-date. I see no
reference to 2025." author's reply " The bibliography has been reordered. In Section 7, I discuss the only relevant
work from the year 2025 that I am aware of this topic. However, the recurring
appearance of an imaginary unit there is difficult to reconcile with the title of this
work. Therefore, I would appreciate specific suggestions for related 2025-papers." The author must have up-to-date references to support the literature review. The author asked me to suggest some references, but according to the journal policy, I am not supposed to give references to cite.

Author Response

1) Author reply most to the comments just "Did" for example, comment 8,9 etc what author did, he didn't explain

Reply Explanation:

The numerical order of the references has been changed.

The application in signal theory was also mentioned in the abstract.

In my opinion comment section must be strong and supporting the study with results.

Reply

The results are presented in Sections 3-6, not in the comment section.

The comments in comment-section 7 are indeed quite strong, as can already be seen from the first sentence of section 7, which highlights a fundamental difference from standard literature.

Reply

The current version of the manuscript contains no inconsistency in punctuation at the aforementioned point.

4) In previous report my comment was "Theorem 5.1: The statement of the theorem must be revised."author's reply "Identifying a specific deficiency would be very helpful here."

In my opinion statement must be clear and support the proof in theorm 5.1 i saw a lack of clarity

Reply

I agree with the reviewer's opinion that statements must be clearly formulated and, as in the present case, supported by appropriate proofs.

Reply

The sentence structure I chose is grammatically correct; in particular, it emphasizes the word "therefore."