The questions below are intended to give you a rough idea of what you can expect on the midterm. The actual midterm will not be of the same length, nor will it have exactly the same types of questions.
Note that you will not be able to use computer algebra during the midterm, so make sure you can do all of these problems by hand!
Find a minimal basis of monomials for each of the following monomial ideals.
Find the \(S\)-polynomial \(S(f,g)\) for each of the following pairs of polynomials \(f,g \in \mathbb{Q}[x,y,z]\) for the lex order.
Determine whether each of the following sets of polynomials are a Gröbner basis in \(\mathbb{Q}[x,y,z]\) for the lex order.
For each of the following ideals in \(\mathbb{Q}[x,y,z]\), find a Gröbner basis in the lex order.
The following sets are Gröbner bases in \(\mathbb{Q}[x,y,z]\) for the lex order. For each set, find the reduced Gröbner basis generating the same ideal.
The set \[ G = \{ xy-z^2,\ xz-y^2,\ y^3-z^3 \} \] is a Gröbner basis for an ideal \(I \subseteq \mathbb{Q}[x,y,z]\) in lex order. Determine which of the following polynomials \(f\) are contained in the ideal \(I\).
Suppose \(I=\langle x^2-y,\ x^2y-z \rangle \subseteq \mathbb{Q}[x,y,z]\). Find bases for the following ideals:
Each of the following is a lex Gröbner basis for an ideal \(I\) in \(\mathbb{C}[x,y,z]\). In each case, describe the set of all partial solutions \((y,z)=(a_2,a_3)\) which extend to full solutions \((a_1,a_2,a_3) \in \mathbb{V}(I)\).