"1. Let r be a positive irrational number. Write x^a y^b < x^c y^d if a+rb < c+rd.
Show that this defines a monomial order."
We need to check that it's a total order (this uses the irrationality), that any decreasing sequence of monomials terminates (this uses the positivity), that it's transitive (really easy), and that a < b implies ac < bc (really easy).
"2. Consider the vector space of 2x4 matrices M, with entries (m_ij), and let X = {M : M's rows are linearly dependent}.
For i < j, both from 1 to 4, let p_ij be the 2x2 determinant using columns i,j of M.
What's the relation of p_ij to X?"
Each p_ij vanishes on X. More specifically, X is the vanishing set of the ideal generated by the six p_ij.
"3. (continuing 2) Lex-order the variables m_11, m_12, m_13, m_14, m_21, m_22, m_23, m_24.
What are the leading terms of the six guys p_ij?"
p_ij = m_1i m_2j - m_1j m_2i. The lex-ordering sees the m_1i term first, if i < j. So the leading term is m_1i m_2j.
"4. (continuing) Show that the p_ij are a Gr\"obner basis, by computing S-pairs."
If the LCM of the two leading terms is just the product, the S-pair quickly reduces to 0.
So the cases to consider are S(p_ij, p_ik), S(p_ik, p_jk), where i < j < k.
(Secretly, I'm thinking about 2xn matrices, not just 2x4, here.)
I like to do this by starting with LCM - LCM, reduce each one using a different generator, get to something less obviously in I, then reduce that until stuck.
Case 1. Start with m_1i m_2j m_2k - m_1i m_2j m_2k,
-> m_2i m_1j m_2k - m_2i m_2j m_1k
-> m_2i m_2j m_1k - m_2i m_2j m_1k
= 0
Case 2. Start with m_1i m_1j m_2k - m_1i m_1j m_2k
-> m_2i m_1j m_1k - m_1i m_2j m_1k
-> m_2i m_1j m_1k - m_2i m_1j m_1k
= 0
