MAS400: Exercises 6

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

1. Show that for a given monomial order ⩽ on R ⩽ F [x_{1}, …, x_{n}] the reduced Gröbner basis of any ideal I ⩽ R is unique with respect to ⩽. (The reduced Gröbner basis of I can depend on the monomial order chosen.)

2. Find an algorithm for reducing a Gröbner basis (for an ideal I of F = [x_{1}, …, x_{n}] w.r.t. some monomial order ⩽) to a reduced Gröbner basis. Write the algorithm in pseudo-code.

3. Let G be a Gröbner basis for the non-zero ideal I ⩽ F [x]. Show that there exists f ∈ F [x] such that f ∈ G and I = ⟨f⟩.

4. Use elimination ideals to determine the following affine varieties.

(a) 𝕍(xy − 1, y^{2} − 1) ⊆ ℂ^{2}.

(b) 𝕍(xy − 1, xz − 1) ⊆ ℂ^{3}.

(c) 𝕍(xy − 1, yz − 1, zx − 1) ⊆ ℂ^{3}.

(d) 𝕍(xy^{2} − 1, xz^{2} − 1) ⊆ ℂ^{3}.

(e) 𝕍(x^{2}y − 1, x^{2}z − 1) ⊆ ℂ^{3}.

(f) 𝕍(x^{2}y − 1, y^{2}z − 1, z^{2}x − 1) ⊆ ℂ^{3}.

(g) 𝕍(x^{3}y − 1, xy^{3} − 1) ⊆ ℂ^{2}