Math 380, Spring 2018, Assignment 5

I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh.

- Thomas Mann, The Magic Mountain

Read:

1. Section 2.2.
2. Section 2.3.

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

1. Partial order (on a set $S$).
2. Total order.
3. Well-ordering.
4. Monomial order.
5. Lexicographic order ("lex").
6. Graded lexicographic order ("grlex").
7. Graded reverse lexicographic order ("grevlex").
8. Leading term (of a polynomial).
9. Leading monomial.
10. Leading coefficient.
11. Multidegree.

Carefully describe the following algorithms:

1. Multivariable division algorithm.

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

1. Theorem relating the well-ordering property with strict descents and weak descents.

Solve the following problems:

1. Section 2.2, problems 1, 2, 3, 10, and 11.
2. Section 2.3, problems 1 and 2.

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