MAS400: Exercises 5

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

1. Fix a monomial order on R = F [x_{1}, …, x_{n}] and let f, f_{1}, …, f_{m} ∈ R with f_{1}, …, f_{m} ≠ 0. Suppose that f rem (f_{1}, …, f_{s}) = 0 for some s ⩽ m. Show that f rem (f_{1}, …, f_{m}) = 0.

2. Let F be your favourite field, and choose your favourite monomial order. State your choices clearly. With respect to your choices find a Gröbner basis for I in each of the following cases.

(a) Ideal I = ⟨xy − 1, y^{2} − 1⟩ in F [x, y].

(b) Ideal I = ⟨xy − 1, xz − 1⟩ in F [x, y, z].

(c) Ideal I = ⟨xy − 1, yz − 1, zx − 1⟩ in F [x, y, z].

(d) Ideal I = ⟨xy^{2} − 1, xz^{2} − 1⟩ in F [x, y, z].

(e) Ideal I = ⟨x^{2}y − 1, x^{2}z − 1⟩ in F [x, y, z].

(f) Ideal I = ⟨x^{2}y − 1, y^{2}z − 1, z^{2}x − 1⟩ in F [x, y, z].

Also find a reduced Gröbner basis for each of the above ideals. NB: Some of these questions may take a long time.