Math 480, Spring 2013, Assignment 5

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

1. Remainder.
2. Monomial ideal.

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

1. Theorem on the multivariable division algorithm.
2. Theorem concerning membership of a polynomial in a monomial ideal.
3. Dickson's Lemma.

Solve the following problmes:

1. Consider the polynomials \(f = x^3 - x^2y - x^2z + x, g_1 = x^2y - z, g_2 = xy - 1.\) Using grlex order, compute the remainder when \(f\) is divided by \((g_1,g_2)\), then when it is divided by \((g_2,g_1)\). Are the remainders the same?
2. Let \(I = \left\langle x^6, x^2y^3, xy^7\right\rangle\subset \mathsf{k}[x, y]\). (a) Draw a picture representing the monomials in \(I\) as lattice points in a plane. (b) If a polynomial is divided by the given generators of \(I\), what terms can appear in the remainder?