Math 480, Spring 2013, Assignment 7

Carefully define the following terms, then give one example and one non-example of each:

1. Normal form (of a given polynomial with respect to a given set of polynomials).
2. \(S\)-polynomial (of two given polynomials).
3. Minimal Groebner basis.
4. Reduced Groebner basis.

Carefully state the following theorems (you need not prove them):

1. Buchberger's \(S\)-pair criterion.
2. Theorem on uniqueness of reduced Groebner bases.

Carefully say how to execute the following algorithms:

1. Buchberger's algorithm.

Solve the following problems:

1. Consider the polynomials \(g_1 = xy^2 - xz + y, g_2 = xy - z^2, g_3 = x - yz^4.\) Show that \((g_1, g_2, g_3)\) is not a Groebner basis with respect to lex order by exhibiting an element of the ideal they generate whose leading term is not divisible by the leading term of any of the given polynomials. (Hint: compute some \(S\)-pairs.)
2. Suppose that \(I = \left\langle g\right\rangle\) is a principal ideal (i.e. generated by a single element). Show that \(\{g\}\) is a Groebner basis for \(I\).
3. Does the \(S\)-pair \(S(f, g)\) depend on which monomial order is used to compute it? Prove your answer.
4. Using Buchberger's algorithm, compute a Groebner basis for the ideal \(I = \left\langle x^2y - 1, xy^2 - x\right\rangle\) with respect to grlex order. Then use your answer to compute a reduced Groebner basis for the same ideal.

Questions

• I am a little bit confused about the Buchberger's algorithm question. I calculated the S polynomial and then divided it by the other values, and at least the first S polynomial was not divisible by either polynomial, so I added it to the basis. Do I then recalculate the S polynomial again and check the remainder by division with the original terms and the term added from the previous step? --Robert.Moray (talk) 09:00, 28 March 2013 (EDT) EDIT: I think I have to test all of the combinations of S polynomials, there are just more combinations than with only two polynomials.

Yes. At least this is what the algorithm specifies. When you do it, however, you will quickly realize that there is actually no need to reconsider the original S-pair. But there will be a bunch of new S-pairs to calculate, which is what makes Buchberger a computational challenge. (N.B.: In this problem I have Buchberger outputting a Groebner basis with four elements. It is non-minimal; after refinement to a minimal G.b. I have two elements - but not the same two elements we started with. So if it's looking more complicated than this, check your calculations.) -Steven.Jackson (talk) 10:35, 28 March 2013 (EDT)

• Professor I am confused about How we went from 2nd baby theory to Grown up theory of general Ideals. Dickson's Lemma says monomial Ideal is generated by finite set of monomials. But when we apply that to grown up theory class notes says By Dickson's lemma, there exists some finite set g1......gs (belongs to) I such that LT(g1).......LT(gs) generates LT(I). I do understand by dickson's lemma LT(I) must be generated by some finite monomials LT(g1).....LT(gs). But I don't understand how did Dickson's lemma said for general ideal there exists some finite polynomial g1.......gs that belongs to a general Ideal? Thank you, Bishesh Baniya

So let \(I\) be an ideal, and choose \(g_1,\dots,g_s\in I\) such that \(LT(I) = \left\langle LT(g_1),\dots,LT(g_s)\right\rangle\). Then \(g_1,\dots,g_s\) generate \(I\). (This is Theorem 4 on page 76; one proves it using the division algorithm.) -Steven.Jackson (talk) 15:19, 12 April 2013 (EDT)