MA3417 Homework assignment 3
Due on November 21, 2013
1. Suppose that < is a monomial ordering on words in {x1 , . . . , xn }. Let us define the recursive path
ordering <rp on words in {x0 , x1 , . . . , xn } as follows:
• If the number of occurrences of x0 in u is less than the number of occurrences of x0 in v, then
u <rp v.
• If there is a tie, that is u and v both have n occurrences of x0 , we have factorisations
u = u0 x0 u1 x0 · · · un−1 x0 un
and v = v0 x0 v1 x0 · · · vn−1 x0 vn
with ui , vi being words that do not contain x0 . Let k = min{i : ui 6= vi }; we put u <rp v if
uk < vk .
Show that <rp is a monomial ordering.
2. (a) Show that the monomials {y i xj } form a basis of the ring Chx, yi/(xy − yx − 1) viewed as a
C-vector space.
(b) Computing products by hand for small i, j, k, l, guess a formula that expands the product of
the basis elements (y i xj )(y k xl ) as a combination of basis elements,
X
i,j,k,l m n
(y i xj )(y k xl ) =
αm,n
y x ,
and prove the formula you obtained.
3. For the ideal (y 2 , x2 y + xyx + yx2 ) ⊂ F hx, yi, compute its reduced Gröbner basis for the DEGLEX
ordering with x > y.
4. Show that there does not exist a monomial ordering for which the ideal (x2 , xy − zx) ⊂ F hx, y, zi
has a finite Gröbner basis.