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

1. 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)