HW #4

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.

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?

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?

4. (continuing) Show that the p_ij are a Gr\"obner basis, by computing S-pairs.