MAS400: Exercises 3

Throughout this sheet F will denote an arbitrary field unless otherwise stated.

1. Perform multivariate division in F [x, y] for the following cases. For critical cases, one may assume that F = ℚ. Do them also with f_{1} and f_{2} interchanged, and for the monomial orders ⩽_{lex} and ⩽_{grlex} in which x > y. (You can also see what happens if x < y if you like.)

(a) f = xy^{3} − xy^{2}, f_{1} = xy − x, f_{2} = y^{3}.

(b) f = x^{2} + xy^{3}, f_{1} = xy + 1, f_{2} = y^{2}.

2. Perform multivariate division in F [x] when f ∈ {x^{4}, x^{4}−1}, f_{1} = x^{3}+1, f_{2} = x^{2} + 1, and also with f_{1} and f_{2} interchanged. Note that there is a unique monomial order in this case. Although F [x] is univariate, we cannot use univariate division as we are trying to simultaneously divide by two polynomials. What is the ideal I = ⟨f_{1}, f_{2}⟩ anyway?

3. Let I = ⟨\underline{x}^{α_{1}}, \underline{x}^{α_{2}}, …, \underline{x}^{α_{s}}⟩ be a (finitely generated) monomial ideal of F [x_{1}, …, x_{n}], and f ∈ F [x_{1}, …, x_{n}]. Show that if we apply multivariate division to divide f by (\underline{x}^{α_{1}}, \underline{x}^{α_{2}}, …, \underline{x}^{α_{s}}) then we get remainder r = 0 if and only if f ∈ I (regardless of the fixed monomial order used).