Math 480, Spring 2014, Assignment 7

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

1. Ideal of relations (among polynomials $f_1,\dots,f_m$) -- this is also called the syzygy ideal.
2. Algebraic relation (among $f_1,\dots,f_m$) -- also called a syzygy.
3. Syzygy (in the sense of astronomy -- use Google, or an actual dictionary, to find this one).

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

1. Theorem expressing the subalgebra generated by $f_1,\dots,f_m$ as a quotient of the polynomial ring $\mathsf{k}[y_1,\dots,y_m]$ (see Proposition 2 in section 7.4).

Describe the following algorithms:

1. Syzygy algorithm (see Proposition 3 in section 7.4).

Solve the following problems:

1. Explain why the word "syzygy" was chosen as a synonym for "algebraic relation."
2. Section 7.4, problems 5 and 6. (These both require the use of some computer algebra system, unless you want to execute Buchberger's algorithm by hand, so they cannot be quiz questions. But they are still worth doing.)

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