Math 380, Spring 2018, Assignment 9

The moving power of mathematical invention is not reasoning but the imagination.

- Augustus de Morgan

Read:

1. Section 2.6.
2. Section 2.7.

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

1. $\overline{f}^{(g_1,\dots,g_s)}$ (the normal form of $f$ modulo the ordered set $(g_1,\dots,g_s)$).
2. Least common multiple (of two monomials).
3. $S(f,g)$ (the syzygy polynomial determined by $f$ and $g$).

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

1. Bound on the multidegree of $S(f,g)$.
2. Cancellation lemma (this is Lemma 2.6.5 in the text).
3. Buchberger's $S$-pair criterion.

Carefully describe the following algorithms:

1. Buchberger's algorithm (to compute a Grőbner basis for a given ideal $\left\langle f_1,\dots,f_s\right\rangle$).

Solve the following problems:

1. Section 2.6, problems 2, 5, 6, and 9.
2. Section 2.7, problem 2(a), using lex order only.
3. (Parameter elimination again) Using lex order in which $t>x>y$, compute a Grőbner basis for the ideal $\left\langle t-x, ty-1\right\rangle$. What does this tell you about the image of the parametrization $x=t,\quad y=1/t$?

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