Math 480, Spring 2013, Assignment 12

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

1. Sylvester matrix (of two polynomials with respect to the variable \(x_1\)).
2. Resultant (of two polynomials with respect to the variable \(x_1\)).

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

1. Theorem on resultants, common factors, and elimination ideals (Proposition 3.6.1 in the text).
2. Theorem relating the evaluation of the resultant to the resultant of the evaluations.
3. Extension theorem.

Do the following problems:

1. Compute the Sylvester matrix and the resultant of \(f = x^2-y^2\) and \(g = x^3-y^3\) with respect to \(x\). Did you expect this result? Why or why not?
2. Compute the Sylvester matrix and the resultant of \(f = x^2+y^2\) and \(g = x^3-y^3\) with respect to \(x\). Did you expect this result? Why or why not?

Questions

1. Should I consider the \(y^i\) as equivalent to \(y^i*1=y^ix^0\)? EDIT: Also, I had another question: So for example with the first problem, there are technically four coefficients for g: 1 for \(x^3\), 0 for \(x^2\), 0 for \(x^1\), and presumably \(-y^3\) for \(x^0\). However, since the largest power of x in f is 2, and in g is 3, I believe this implies that Syl(f,g,x) should have 2 columns in g, 3 columns in f, and to have a calculable determinant, should be 5x5. But since g's coefficients are only represented in two columns in the Syl(f,g,x) matrix, and g has four coefficients, does this mean that the coefficient for \(x^3\) will disappear? In general, can multiple coefficients disappear in this way? --Robert.Moray (talk) 00:50, 2 May 2013 (EDT)