Math 380, Spring 2018, Assignment 8

Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.

- John Locke, Second Reply to the Bishop of Worcester

Read:

1. Section 2.4.
2. Section 2.5.

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

1. $\left\langle LT(I)\right\rangle$ (the leading term ideal of an ideal $I$).
2. Grőbner basis (for an ideal $I$).

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

1. Dickson's Lemma.
2. Ascending chain condition for monomial ideals.
3. Hilbert basis theorem.
4. Theorem concerning the existence of Grőbner bases.
5. Ascending chain condition for arbitrary ideals.

Carefully describe the following algorithms:

1. Algorithm to decide whether $f\in\left\langle g_1,\dots,g_s\right\rangle$ when $g_1,\dots,g_s$ form a Grőbner basis for the ideal that they generate.

Solve the following problems:

1. Section 2.5, problems 1, 7, 8, 10, 17, and 18.

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