MA341D Homework assignment 3
Due on March 10, 2016
Note that there will be no classes on February 23 and February 25; we shall have replacement
classes for those in the last week of the term.
1. In Question 2 of the previous homework, you made a conjecture about the reduced Gröbner
basis (for the glex order with x < y) for the ideal generated by the noncommutative
polynomial f = xyx − yxy. Using Diamond Lemma, prove your conjecture.
2. In Question 3 of the previous homework, you investigated the reduced Gröbner basis
(for the glex order with x < y < z) for the ideal generated by the noncommutative
polynomials f = xz − zx, g = xyx − yz, and h = y 2 x. Write down the reduced monomials
of length up to 5 with respect to the truncation of the Gröbner basis that you computed.
[Bonus question (not assessed): using these normal monomials, guess what the full reduced
Gröbner basis is, and prove your conjecture.]
3. Pick some order of noncommutative monomials in x, y, z and compute (by hand, without
computer algebra software) the reduced Gröbner basis for the ideal
(x2 + yz, x2 + 3zy) ⊂ F hx, y, zi.
Describe the reduced monomials with respect to the Gröbner basis you computed.
4. Pick some order of noncommutative monomials in x, y and compute (by hand, without
computer algebra software) the reduced Gröbner basis for the ideal
(x2 y − 2xyx + yx2 + y, xy 2 − 2yxy + y 2 x + x) ⊂ F hx, yi.
Describe the reduced monomials with respect to the Gröbner basis you computed.
5. (a) Show that the system of (commutative) polynomial equations


xz = y + 1,




yt = z + 1,
zu = t + 1,



tx = u + 1,




uy = x + 1
has infinitely many solutions over C.
n
(b) Consider the recurrence relation an+1 = 1+a
an−1 with some given a0 , a1 . Show that if
for all n the element an is well defined (which happens if the sequence has no zero terms),
then we have ak+5 = ak for all k. How is this result related to your computation in the
first part of this question?