Math 380, Spring 2018, Assignment 10

The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.

- Saint Augustine

Read:

1. Section 2.8.

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

1. Minimal Grőbner basis.
2. Reduced Grőbner basis.
3. Elimination ideal.

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

1. Theorem concerning the uniqueness of reduced Gröbner bases.

Carefully describe the following algorithms:

1. Minimization algorithm.
2. Reduction algorithm.
3. Algorithm to decide when one ideal is contained in another.
4. Algorithm to decide when one ideal equals another.

Solve the following problems:

1. Section 2.8, problems 1, and 3.
2. (Optional; you may need a computer algebra system) Section 2.8, problems 5 and 10.
3. Consider the system of linear equations $$\begin{align*}x+y+z&=1\\x+2y\quad\ &=2\\y-3z&=3.\end{align*}$$ First, solve the system using Gauss-Jordan elimination. Then, compute a reduced Grőbner basis (with respect to lex order) for the ideal associated with the system. How are your two calculations related?

--------------------End of assignment--------------------