The Poincaré Index and Its Applications

Round 1

Reviewer 1 Report

This is an interesting but imperfect paper. It could be publishable after significant revisions.

I need to make it clear upfront that, while I consider myself a competent mathematician, I am by no means an expert on the subject matter of this paper. If you are able to send it to an expert for a second opinion, you should.

The question of explicitly computing the index of a vector field is an important one, and the author presents an interesting approach. However, too much context is given for the very broad subject, and too little for the specific technique used.

The introduction is very leisurely. I am not sure what Universe expects, but the first 5 sections (pages 1-6) are expository, written at an undergraduate level. It might be more efficient to abbreviate these to just a page or two setting up the problem and the homological approach to it.

The heart of the matter is in sections 6-8, where the author describes his approach, first in general and then in the useful quasihomogeneous case. Considering that the index of a vector field is an old and much studied subject, as are homological techniques, it seems somewhat surprising that there are many references to generic works, 8 references to previous technical works by the author himself, but essentially no other references to current mathematics. (There are a couple of relevant references to works of Gomez-Mont, but those are from the 1990s.) The subject does have a large literature. For example, several well-known works on the subject are listed in the Math Overflow note: https://mathoverflow.net/questions/379718/how-to-compute-the-index-of-a-vectorfield-defined-by-analytic-formula.

If none of these is relevant to the work at hand, perhaps the author could explain the advantages of his own methods compared to existing literature.

For example, is it not possible to work out the examples in section 7 using existing technology?

To my mind, the “applications” section 9 is unacceptable. It consists of a list of subjects where the Poincare index is relevant, with no specific connection to the results of the present paper. Instead of referring aimlessly to such a wide range of topics, it would be much better to give one or two actual instances where the results of this paper solve a pre-existing problem.

So, to summarize: a flawed presentation of good work on an important problem. Well worth publication after a thorough rewriting.

Author Response

Response to Reviewer 1 Comments

Point 1: I need to make it clear upfront that, while I consider myself a competent mathematician, I am by no means an expert on the subject matter of this paper. If you are able to send it to an expert for a second opinion, you should.

Response 1: All published author's articles containing different parts of the described approach have already been reviewed by many recognized experts in the field. I strongly recommend that Anonymous Reviewer 1 read or at least take a look at articles from the bibliography (e.g., [21]) instead of reading Internet networks.

Point 2: The question of explicitly computing the index of a vector field is an important one, and the author presents an interesting approach. However, too much context is given for the very broad subject, and too little for the specific technique used.

Response 2: “Too much context” is used in order to introduce readers in the problem but they do not need to understand any specific technique because the final results are formulated in elementary terms.

It is not difficult to see that the described approach is elementary; it does not require any special knowledge and specific technique. You only need to understand standard operations on integers (such as 2x2=4) and the binomial formulas (such as (x+y)².= x²+ 2xy+y²).

For completeness, it should be worthy underlined also that the meaning of the equality 2+2=4 was explained in detail by H. Poincaré in his famous book [2], and everybody can find the binomial formula (which dates back to the era of I. Newton) in many textbooks for schoolchildren (it is sometimes called the Newton binomial formula).

Point 3: The introduction is very leisurely. I am not sure what Universe expects, but the first 5 sections (pages 1-6) are expository, written at an undergraduate level. It might be more efficient to abbreviate these to just a page or two setting up the problem and the homological approach to it.

Response 3: On my opinion, it is a very important and fruitful idea to publish such an article in the special issue of Universe in order to draw the attention of potential readers (including competent mathematicians) to new modern methods (see also the section “Conclusion of the author” at the end of this Response).

The goal of the first 5 sections (pages 1-6), “written at an undergraduate level”, is to introduce in the subject a wide range of potential readers. Therefore I shortly describe some details of the previous history and several simple well known methods of calculation of the index. I did not see elsewhere such a collection of facts and conclusions.

The presented method in a certain sense partially implements a very general idea of Poincaré, who created foundations of the qualitative theory of differential equations (see Introduction). In other words, it is about how to describe the basic properties of solutions to differential equations without solving them (using only their representations).

Moreover, the described approach is aimed at introducing a wide range of hypothetical readers of any level and competence into the topic, including students, undergraduates, masters, scholars, mathematicians, physicists, etc. That is why the expository sections play a key role; they contain a summary and a brief description of the simplest methods due to Poincaré and his famous followers.

In fact, the methods mentioned in sections 1-4 are not so simple, because they are based on a number of very non-trivial ideas, concepts and tools (such as homotopy, integration, differentiation) from topology, analysis, geometry, etc. Of course, there are many other methods, much more complicated (sometimes very cumbersome and too sophisticated). Their description is easy to find in many textbooks, articles, book collections and Internet networks. Therefore, it makes no sense to reproduce relevant materials in the article.

Point 4: Considering that the index of a vector field is an old and much studied subject, as are homological techniques, it seems somewhat surprising that there are many references to generic works, 8 references to previous technical works by the author himself, but essentially no other references to current mathematics. (There are a couple of relevant references to works of Gomez-Mont, but those are from the 1990s.)

Response 4: This topic is indeed very “old” and has been studied in many works by many authors. However, to develop a new method, it is necessary to have a good understanding of the main ideas of all predecessors.

It seems to be enough to understand the definition of the homological index for vector fields with the use of “a couple of relevant references to works of Gómez-Mont”. However, the computation of homological index is a completely different problem solved by the autor’s in the cited works. For a more detail explanation of the author’s results, an additional reference [24] relating to the calculations in Example 2, using the method of Gómez-Mont with collaborators, is included to the list of references.

Point 5: The subject does have a large literature. For example, several well-known works on the subject are listed in the Math Overflow note: https://mathoverflow.net/questions /379718/how-to-compute-the-index-of-a-vectorfield-defined-by-analytic-formula. If none of these is relevant to the work at hand, perhaps the author could explain the advantages of his own methods compared to existing literature.

Response 5: The present work is not a survey of known literature. There are many other people interested in writing such kind papers, books, etc. The methods mentioned in Math Overflow much more complicated than the presented one. Moreover, all other methods are based essentially on a number of very non-trivial ideas, concepts and tools from algebra, geometry, topology, analysis such as spectral sequences, hypercohomology, resolutions of coherent sheaves, integration, differentiation and many others. On the contrary, the author's method is elementary.

Point 6: For example, is it not possible to work out the examples in section 7 using existing technology?

Response 6: The section 6 contains the most important definition of the logarithmic index. As the author remarks in [20] “the systematic use of the theory of logarithmic forms permits one not only to simplify the calculations of the homological index dramatically but also to clarify the meaning of the basic constructions underlying many papers on the subject”.

Thus, Example 1 is considered in the last section of Arnold's well-known textbook [11], where the calculations are based on the classical theory of ordinary differential equations, described on the previous two hundred pages.

The next Example 2 is analyzed by Gómez-Mont with collaborators in a very complicated work [24], where they constructed a cumbersome multidimensional complex of coherent sheaves, the picture of which takes up almost 2 pages. Then they use a very powerful computer, a special computer system of algebraic computations and procedures (not published) and so on. As a result, they state that the value of index is equal to 12. The author’s original method gives this result immediately: 12 = 1+4+4-2-2+4+3. To clear things up, I recommend to the Anonymous Reviewer 1 to take a look at the reference [23].

It is not possible to analyze Example 3 with the use of "old" or "known" methods described anywhere (e.g. in Math Overflow, by Gómez-Mont, etc.). Indeed, these equations define the simplest singularity in three-dimensional space; the corresponding example goes back to F.S. Macaulay (see his famous book "The Algebraic Theory of Modular Systems" (1916), p.53, example (i)). There are many areas of modern algebra, geometry and topology where this singularity plays a key role. For example, the concept of Cohen-Macaulay rings, modules, singularities were inspired by this example (I hope that there are many posts on Math Overflow network concerning these subjects).

Point 7: To my mind, the “applications” section 9 is unacceptable. It consists of a list of subjects where the Poincaré index is relevant, with no specific connection to the results of the present paper. Instead of referring aimlessly to such a wide range of topics, it would be much better to give one or two actual instances where the results of this paper solve a pre-existing problem.

Response 7: The section 9 “applications” is primarily addressed to those who are interested in developing the theory in other directions or are able to exploit these ideas in other settings (such as Poisson varieties --  see e.g. [12], [26]).

Point 8: So, to summarize: a flawed presentation of good work on an important problem. Well worth publication after a thorough rewriting.

Response 8: No comments.

Author's final comments

Dear Anonymous Reviewer 1,

thank you very much for your attention and remarks. However, I am afraid that if I follow all your advice, the article will turn into a short note containing only 3-4 formulas, some notations and definitions. As a result, it will be only suitable for reference books, handbooks, tutorials, etc. (such as Herkimer's Handbook of Indispensable Information, which is mentioned in the famous humorous story “The Handbook of Hymen” by the classic of American literature O'Henry; it is available at https://bingebooks.com/book/ the-handbook-of-hymen).

However, in my opinion, the purpose of the Special Issue «Singularities in Space-time» of the journal «Universe» is quite different.

Author Response File: Author Response.pdf

Reviewer 2 Report

I am extremely perplex. This seems to me an old work, just marginally rearranged. The author should clarify: (i) His present hosting Institution. (ii) Which part of his conclusions are unpublished original work.

Author Response

Response to Reviewer 2 Comments

Point 1: I am extremely perplex (perplexed ?).

Response 1: If I understand your remark correctly, then you are not alone (as far as I know).

Unlike the well known methods, the described approach is absolutely elementary; it does not require any special knowledge in topology, analysis, geometry and special tools, such as integration or differentiation. Readers only need to understand standard arithmetic operations on integers (such as 2x2=4) and the binomial formulas (such as (x+y)².= x²+ 2xy+y²).

Moreover, in a certain sense the described method partially implements the idea of Poincaré, who created the foundation of the qualitative theory of differential equations (see Introduction). In other words, it is about how to describe the basic properties of solutions to differential equations without solving them (using only their representations).

Point 2: This seems to me an old work, just marginally rearranged.

Response 2: This article is not as old as the works of Poincaré, although some of them have also been revised many times by many authors, such as A.Andronov (Poincar´e’s concept of the limit cycle), A.Einstein  (the general theory of relativity), etc., but they are still relevant today and very important for those who are interested in scientific work.

In contrast, in the present article the author describes his own ideas and methods developed over the past few years such as the theory of logarithmic differential forms, the theory of quasi-homogeneous complete intersections, the theory of logarithmic index of vector fields and differential forms and so on.

Point 3: The author should clarify: (i) His present hosting Institution. (ii) Which part of his conclusions are (is ?) unpublished original work.

Response 3: The answer to the item (i) is disposed under the title of the article, while partial answers to the comment (ii) are given in the section “Applications”.

Author Response File: Author Response.pdf