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W Mon Tue Wed Thu Fri
1  
 
1.1, 1.2
 
1.3, 1.4
Autobio
1.5, 2.1
 
2.2, 2.3
 
🍂 Fall Break 🍂
2 2.4, WR1
 
2.5
Batch A
2.6, (2.9)
 
2.7, (2.10)
 
2.8
Batch B
3 3.1, 3.2, WR2
 
3.3
 
3.4, (3.5)
 
4.1
Batch C
4.2
 
4 WR3
Batch D
 
 
 
Batch E, FR

How to read this calendar:

Sections

1.1: Multivariable polynomials; Affine space

CC: 5
Easier: 2
Harder: 3, 4, 6

Notes:

1.2: Affine varieties

CC: 1, 8
Easier: 3, 4, 5, 6
Harder: 2, 7, 9, 13+14, 15

Notes:

1.3: Parametrization

CC: 1
Easier: 2, 3, 12
Harder: 4, 5(b-d), 6, 7, 8, 13, 14+15

Notes:

1.4: Ideals to Affine Varieties and Back

CC: 2, 3
Easier: 4, 9, 14, 16
Harder: 6, 7+8, 10, 13, 15, 17+18, 19

Notes:

1.5: Polynomials in One Varaible

CC: 1, 5
Easier: 4
Harder: 2, 7, 8, 10, 12, 14+15, 16+17

2.1: Introduction to Gröbner Bases

CC: 1
Easier: 2, 3
Harder: 4, 5

Notes:

2.2: Monomial Orderings

CC: A, B, 1
Easier: 2, 3, 4, 6, 8
Harder: 9+5, 7, 10, 11+12, 13

Lettered problems:

  1. Make a list of all of the monomials in \(k[x,y]\) in increasing order using lex with \(x > y\). You can use “…” in your list, but make sure you identify a clear pattern.
  2. In \(k[x,y,z]\), there are 6 monomials of total degree 2. Rank these 6 monomials in descending order using grlex. Then rank them in descending order again using grevlex.

Notes:

2.3: Multivariable Division Algorithm

CC: 1, 2
Easier: 4, 5, 9
Harder: 6+7, 8, 10, 11

Notes:

2.4: Monomial Ideals

CC: 1, 2, 3
Easier: 4(b), 5, 6, 9, A
Harder: 7, 8, 10, 11

Lettered problems:

  1. Suppose \(I \subseteq k[x_1, \dotsc, x_n]\) is a monomial ideal. Describe the corresponding affine variety \(V(I)\) geometrically.

Notes:

2.5: Gröbner Bases

CC: 1, 4, 14
Easier: 5, 7, 10, 12, 15, 16, 17, A
Harder: 3, 6, 8, 13, 18

Lettered problems:
  1. We know that any ideal \(I\) in the single variable polynomial ring \(k[x]\) is of the form \(I = \langle h \rangle\) for some \(h \in k[x]\). Suppose \(G = \{g_1, \dotsc, g_t \}\) is a subset of \(I\). When is \(G\) a Gröbner basis for \(I\)? Make a nontrivial statement of the form “\(G\) is a Gröbner basis for \(I\) if and only if…,” and then prove it.

Notes:

2.6: Properties of Gröbner Bases

CC: 1, 2, 5
Easier: 3, 4, 9, 10, A
Harder: 6, 7, 11, 8+12, 13

Lettered problems:

  1. For any \(f, g \in k[x_1, \dotsc, x_n]\) and \(a, b \in k\) nonzero, prove that \(S(f,g) = S(af, bg)\).

Notes:

2.7: Buchberger’s Algorithm

CC: 1, 2, 3
Easier: 6, 8, 9
Harder: 7, A, 10(b-c), 11, 13, 14, A, B

Lettered problems:
  1. Let \(A = (a_{i,j})\) be a \(n \times m\) matrix. Row \(i\) of \(A\) defines a linear polynomial \(f_{A,i} = a_{i,1} x_1 + \dotsb + a_{i,m} x_m\) in \(k[x_1, \dotsc, x_m]\). Let \(I_A = \langle f_{A,1}, \dotsc, f_{A,n} \rangle\) be the ideal generated by these linear polynomials.
    1. Let \(B\) be a \(n \times n\) matrix. Show that \(I_{BA} \subseteq I_A\).
    2. Suppose \(B\) is an invertible \(n \times n\) matrix. Show that \(I_{BA} = I_A\).
    3. Use part (b) to solve problem 10(a) in the textbook.
  2. Write two Sage functions: buchberger and reduce_groebner.
    • The function buchberger should implement the algorithm described in theorem 2. It should take as input a polynomial ring R and a list F of polynomials in R, and output a list that is a Gröbner basis for the ideal generated by the polynomials in F.
    • The function reduce_groebner should take as input a ring R and a list of polynomials G that is a Gröbner basis for the ideal generated by the polynomials in G. It should output the unique reduced Gröbner basis for that same ideal. You can assume that G is guaranteed to be a Gröbner basis; in other words, it’s okay if your function reduce_groebner does something unexpected when G is not actually a Gröbner basis.

    Your functions can make use of the functions quo_rem_list and S in the Sage reference, but nothing else (as in, if you want to have a function that does something else, you should write it yourself). If you’re unsure if using some particular function is legitimate, ask me!

    This is mostly a coding problem. In the PDF you submit through Gradescope, just write a blurb saying you did this problem, and then send me an email attaching plaintext .sage files containing the code for your functions.

Notes:

2.8: Applications of Gröbner Bases

CC: 1, 3
Easier: 2, 5, 6
Harder: 7, 11, A

Lettered problems:

  1. In the setup of exercise 11, I’ve done some computations that suggest that, for every positive integer \(n\), there exists a rational number \(r(n)\) such that the three given equations imply \(a^n + b^n + c^n = r(n)\). But I don’t know how to prove this. Can you prove this? Alternatively, can you find an integer \(n\) for which this is not true?

Notes:

2.9: Refinements of the Buchberger Criterion

Easier: 1, 3
Harder: 2

Notes:

3.1: Elimination and Extension

CC: 1, 2(a-c), 5
Easier: 3, 4, 6, 8
Harder: 7, 9

Notes:

3.2: Geometry of Elimination

CC: 2
Easier: 1
Harder: 4, 5+A

Lettered problems:
  1. Give an example (with proof) of an ideal \(I \subseteq \mathbb{C}[x,y]\) such that \(\pi_1(V(I)) \subsetneq V(I_1)\), where \(\pi_1 : \mathbb{C}^2 \to \mathbb{C}\) is projection onto the \(y\)-axis and \(I_1 = \mathbb{C}[y] \cap I\) is the first elimination ideal. Possible hint. By problem 5, any such ideal \(I\) must have the property that \(I_1 = \{0\}\). What does this tell you about \(V(I_1)\)? And what does this plus the closure theorem tell you about \(\pi_1(V(I))\)? It’s helpful to use this geometric intuition to construct an appropriate ideal \(I\), but be careful: you’re working over \(\mathbb{C}\), not \(\mathbb{R}\).

3.3: Implicitization

CC: 1, 6(a,b)
Easier: 2, 4, 8, 14
Harder: 3, 7, 10, 11, 12

Notes:

3.4: Singular Points on Curves

CC: 1, 8
Easier: 2, 3, 5+6
Harder: 9, 10+11, 12

Notes:

3.5: Proof of Extension Theorem

Easier: A, 1, 4

Lettered problems:
  1. Let \(\varphi : A \to B\) be a surjective ring homomorphism.
    1. If \(I \subseteq A\) is an ideal, show that \(\varphi(I)\) is an ideal as well.
    2. Suppose \(I = \langle a_1, \dotsc, a_n \rangle\) for some \(a_1, \dotsc, a_n \in A\). Show that \(\varphi(I) = \langle \phi(a_1), \dotsc, \phi(a_n) \rangle\).
    3. Explain how problems 2 and 5(a) in the textbook follow from parts (a) and (b) above.

Notes:

4.1: Nullstellensatz

CC: 2, 3
Easier: 1, 4
Harder: 7, 8, 9, 10

Notes:

4.2: Ideal-Variety Correspondence

CC: 1, 2, 14
Easier: 3, 4, 5, 7, 8, 12, 13
Harder: 6, 9+10, 15, 16+17

Notes: