Arnold’s Piecewise Linear Filtrations, Analogues of Stanley–Reisner Rings and Simplicial Newton Polyhedra
Round 1
Reviewer 1 Report (New Reviewer)
Dear Professor,
Kindly see the attached report.
Comments for author File: 
Comments.pdf
Author Response
The author is grateful to the reviewer for the work on the analysis of the article and the positive assessment made. The reviewer's report does not suggest any revisions to the article.
Reviewer 2 Report (Previous Reviewer 1)
No modification required.
I strongly support the acceptation of the paper, written by a Master and pioneer on this topic. I read the Russian version (at the beginning of August) and the English version will have more readers.
I repeat what I wrote:
`` The long paper (but length is not a problem, at least on-line) contains several new results and different constructive proofs of some of his old results.
The History part is a gem, historical details, the explanation of the new parts and a list of open areas (section 1.4) for research with at each of the area a claim (I and/or my students plan/do not plan to work on this) and it end with a brief description of arXiv:2006.11795. ''
Author Response
The author is grateful to the reviewer for the work on the analysis of the article and the positive assessment made. The reviewer's report does not suggest any revisions to the article.
Reviewer 3 Report (New Reviewer)
The paper presents a classical result of the author (published in 1974 in Inventiones Math.) regarding the codimension of an ideal generated by generic Laurent polynomials. The novelty consist in presenting more details regarding the proof of the aforementioned result, in certain particular cases. The paper is well written and deserves to be published after certain (minor) issues are tackled, as:
- Write Cohen-Macaulay instead of Kohen-Macoley (in keywords)
- Write Shelling instead of Schelling (I know that in french is "schelling" but the same terminology should be used all over the paper)
- Use \left( and \right) in formulas, page 6.
- Please avoid citing wikipedia! Cite instead a basic book on the respective domain.
In general, please read carefully your paper in order to find other misspelling or misprints.
Author Response
The author is grateful to the reviewer for the work on the analysis of the article and the positive assessment made. The reviewer suggested correcting a number of typographical errors in the article and making a number of editorial changes. Here is a report on the fulfillment of these requirements.
- Write Cohen-Macaulay instead of Kohen-Macoley (in keywords)
Done.
- Write Shelling instead of Schelling (I know that in french is "schelling" but the same terminology should be used all over the paper)
Done. Found 5 misprints of this type. All of them have been corrected.
- Use \left( and \right) in formulas, page 6.
Done. 4 more situations were found in which this remark is applicable.
- Please avoid citing wikipedia! Cite instead a basic book on the respective domain.
Done.
- In general, please read carefully your paper in order to find other misspelling or misprints.
Did my best. The author spent several hours fulfilling this requirement and corrected about a dozen typos. In addition, the author has paid for the English Editing Service and plans to reproof the manuscript upon receipt of the English Editing result.
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
Comments for author File: Comments.pdf
