# Talk:Groebner basis

Dear Dr. Buchberger and Dr. Kauers,

You have written a delightful article on Gr\"obner Bases.

I noticed some misprints and I have a few suggestions for improvements. I will number these in case you have questions or need additional explanations.

For the review process, I printed your article. So, the page numbers refer to a paper version.

1. Would it be possible to put the reference in numerical order?

For example, on page 1: [2,3,4,7,9,......, 47, 48]. This comment applies to other places in the text also.

2. The basis, {xz^2 ......-5z} at the top of page 3 could be replaced by

B = {xz^2 ....... -5z}

3. In Fig. 1, x^2 y^3 should be x^3 y^2

Same change at the bottom of page 3.

4. On page 4, again B = {xz^2 .......... -5z}

5. The reduction on page 5 is fine but there are small errors in the explanation.

Similarly, the second and third arrow are obtain using the first (NOT third) and the second (NOT third) basis element.

I think the reader will find the reduction easier to follow if you also list the multipliers that were used.

In the first reduction you indeed used the second basis element with multiplier 3xz In the second reduction you used the first basis element with multiplier x In the third reduction you used the third basis element with multiplier 9z.

6. Does the arrow have two slightly different meanings? Obviously, it stands for reduction and

also for the binary relation f \rightarrow g. That is a minor point.

7. If Scholarpedia allows you to use equation numbers than you could avoid having to give the

basis B explicitly FIVE TIMES in the first five pages of the article (notice the repitition: B occurs twice on page 3, once on page 4, once on page 5, and again on page 6). You should tell the guys at Scholarpedia that equations numbers were invented to avoid such repetitions!

8. On page 6, again you could add the multipliers to make matters easier to follows.

In the first group of reductions on page 6, you used multipliers x, 3xz, and 6z.

In the second set of reductions, you used 3xz, 9z, x, and 3z.

In the third set of reductions, you used yz^2, xy, 3yz, -5y, and 5 z^2

To get the successor of g, which is 25z^3 - 5z, you used the third basis element with multiplier -1. (It would not hurt to add that).

9. Still on page 6; editing: It is not hard to see that both definitions are equivalent (and to ....).

10. Page 7: ... a root of this univariate polynomial of degree 7.

11. Why not use the standard \prec symbol instead of < for lexicographic order?

12. On page 8 (under canonicality property): does ~ stand for the equivalence relation (and congruent) also?

13. On page 9, editing: ... many possible linear combinations (NOT combination) will turn h INTO (not to) f.

If B were (NOT was) ....

14. I printed your article and the explanation of the reductions near the middle of page 9 does NOT print

properly (text ran off the right edge). Readers will want to print you article! So, make sure it prints properly.

15. Here are the multipliers used in that reduction: z^2, 4x, -12, -2, 16z, and -8 (in case you like to add them).

16. Bottom line of page 9: perhaps replace y + 2 by 2 + y.

17. Bottom line of page 9: perhaps replace y + 2 by 2 + y (also on top of page 10). [Reason: consistency with

2 + y at bottom of page 10].

18. Again a printing problem on pages 10 and 11: text ran over the edge of page on right.

19. In many places SPOL is in capital letters, but you use spol (lower cases) on page 11 (3 places).

20. Last line of page 11: ... in all computer algebra systems like ...

21. On page 12: Rephrase: ... in commutative algebra, the field in which Gr\"obner bases theory originated.

22. On page 14: did you mean equations and inequalities? [You wrote: equations and inequations???]

23. On page 14: even more simple = even simpler.

24. On page 15: no period after the fourth equation since the sentence continues with the words "hold at the same time."

I noticed a few more errors in punctuations in other places. Please check.

25. Printing problem with page 16.

26. Page 17, middle of page: ... the integral cancels the derivative (the "with" is not needed).

27. Page 18, why not use a single notation: either p(0) or p_0, etc.

28. Page 19: Fig. 9 should be Figure 9 (2 places).

29. Page 20, The numbering of the p_1 through p_6 is incorrect (you use p_4 twice) and there are SEVEN equations in total.

For the same reason: p_1 = ..... = p_7 = 0.

30. Page 20, just above box with code: .... the product of TWO (not to) integers.

31. Page 22. No period after (i = 1,....,m).

32. The bibliography needs a bit of editing to end up with similar looking references in consistent style.

Do you want the add the editors (A.M. Cohen, H. Cuypers, and H. Sterk) to [14]?

33. Do you want to define EUROCAM and LNCS? Do you want to define SYNASC'05 (since you define what ISSAC stands for)?

34. Ref. [39]: remove left over latex symbols.

35. Do you want to define POPL'04 (since you define ISSAC)?

Apart from the above remarks, this is a great educational article. The algorithms are well-explained and the applications are interesting. It is a pity that the article is in some version of html instead of LaTex.

Willy Hereman (whereman@mines.edu), Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, CO 80401-1887, U.S.A.

## Contents |

## Reviewer A:

Dr. Buchberger and Dr. Kauers,

I have submitted my referee report.

Willy Hereman

## Author Mkauers : Response to Referee A

Dear Prof. Hereman!

Thanks a lot for your careful reading and the helpful suggestions. It is technically not possible to ensure that the text will print well, but we tried to at least shorten some long lines in display equations.

Only for four of your remarks, we did not change the text. Let me try to justify, using the numbering in your listing as reference.

6. By \rightarrow we always mean the binary relation. Note that this does not prevent us from using the arrow also in the context of a reduction, because this is a special case of the binary relation. We prefer not to address this point in the article because it will most likely produce more confusion than clarity among the nonexpert readers.

7. The repetition of the basis is intentional because it is easier to check the reduction steps if a copy of the basis elements is written near. We thought of readers who don't print the article and would have to scroll back and forth.

22. We did mean inequations (A\neq B), not inequalities (A** -5y <---- , and 5 z^2 (notice that you wrote 5y).
**

**2. In the two lines that follow the third set of reductions:
**

.... (f cannot ... at all. Using the third basis element with multiplier -1, the only successor of g is 25 z^3 - 5 z \ne f.)

3. In the application to Coding Theory:

p_1 should be replaced by p_2 in the third equation of the algebraic system that determines the unknown quantities e_1, e_2, ....., e_5 and a_0, a_1, ...., a_n.

I certainly agree with the rest of you comments about my referee report.

Please let me know when the corrections are made so that I can push the "accept" button.

Best wishes,

Willy Hereman

Boulder, October 11, 2010

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The review editor started messing thing up by cutting of part of my text and inserting items in the wrong place. So here is my message again:

Dear Dr. Kauers and Dr. Buchberger,

The revised version of the article is pretty much ready to be accepted.

I found two small misprints (would be great if these could be corrected) and I have one more suggestion.

1. In the third set of reductions (on page six, 10 lines above "Fundamental Properties of Gr\"obner Bases"),

you actually used the multipliers: y z^2, xy, 3 yz, ---> -5y <---- , and 5 z^2 (notice that you wrote 5y).

2. In the two lines that follow the third set of reductions:

.... (f cannot ... at all. Using the third basis element with multiplier -1, the only successor of g is 25 z^3 - 5 z \ne f.)

3. In the application to Coding Theory:

p_1 should be replaced by p_2 in the third equation of the algebraic system that determines the unknown quantities e_1, e_2, ....., e_5 and a_0, a_1, ...., a_n.

I certainly agree with the rest of you comments about my referee report.

Please let me know when the corrections are made so that I can push the "accept" button.

Best wishes,

Willy Hereman

Boulder, October 11, 2010

# ==

Points 2 and 3 are fixed as requested. Thanks. As for Point 1, I think the multiplier 5y is correct: In order to get from 5xyz^2-xy back to -xy+15y^2z-5y, you have to subtract the 5y-fold of xy^2-3yz+1, not the (-5y)-fold.

Best wishes, Manuel

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Dear Manual,

As of point 1, you are correct! Late at night, the signs get blurry and my elementary mathematics skills deteriorate.

I will now hit the button to accept the paper.

You provided a great service the the scientific community by making this article available on Scholarpedia.

With kind regards,

Willy

Willy Hereman Golden, October 12, 2010.

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## Possible error in the section The Canonicality Property

I have a question about the examples used in the Canonicality Property section. I did the reduction for the polynomials \(f\) and \(g\) by hand and got \( 2z^2+\tfrac34x-\tfrac{11}2\text{ instead of }2z^2+\tfrac34x+\tfrac{11}2 \) for both polynomials. I'm just learning about Gröbner bases, and I'm not sure if this makes a difference in this case or if I made an error along the way. Just thought I'd pass my observation along and see if I could get some more insight into Gröbner bases or polynomial division.