Math 480, Spring 2013, Assignment 12
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Carefully define the following terms, then give one example and one non-example of each:
- Sylvester matrix (of two polynomials with respect to the variable \(x_1\)).
- Resultant (of two polynomials with respect to the variable \(x_1\)).
Carefully state the following theorems (you do not need to prove them):
- Theorem on resultants, common factors, and elimination ideals (Proposition 3.6.1 in the text).
- Theorem relating the evaluation of the resultant to the resultant of the evaluations.
- Extension theorem.
Do the following problems:
- Compute the resultant of \(f = x^2-y^2\) and \(g = x^3-y^3\) with respect to \(x_1\). Did you expect this result? Why or why not?
- Compute the resultant of \(f = x^2+y^2\) and \(g = x^3-y^3\) with respect to \(x_1\). Did you expect this result? Why or why not?
