# Live events

• Group sessions:
• Wed 10am MT: Jules, Alayna, Elizabeth
• Wed 11am MT: Dominic, Oliver, Marston
• Notes: 11/18, 12/2, 12/9
• Office hours:
• Mornings: Mon Fri 10-11am MT, Tue Thu 11-12am MT
• Afternoons: Mon Tue Thu Fri 2-3pm MT
• More by request; just email and ask!
• Notes: 11/30, 12/4, 12/7, 12/10, 12/14

# Calendar

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

• Things above the line are due at 8am MT on that day, and things below the line are due at noon MT on that day.
• Abbreviations:
• Batches:
• Batch A = 1.1-5, 2.1
• Batch B: 2.2-4
• Batch C: 2.5-8, (2.9)
• Batch D: 3.1-2
• Batch E: 3.3-4, (3.5), 4.1-2

# Sections

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:

• For the problems that ask you make a sketch, don’t just plug in numbers, plot points, and interpolate. Also, don’t just get a computer to make sketches for you. Instead, try to manipulate equations by hand and reason about what the plot will look like. Building up geometric intuition is very important!
• Problems that are conjoined with a “+”, like “13+14” above, count as one problem.
• Video and 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

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:

• Problem 9 is the reason that problem 5 is harder than problem 4. Doing problem 9 first will help you avoid a potential mistake in problem 5.
• One can use a vaguely similar strategy for both problems 10 and 5.
• Video and 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:

• It’s important to understand the statements of lemma 5 and theorem 6. The proofs of these two results are fairly confusing; don’t stress too much about them.
• Problem 5(d) should say $$g = z^2 - 3z$$ (S-polynomials only make sense for polynomial functions!). This typo was corrected in the 2018 printing.
• If you do “8+12,” you might find it convenient to do problem 12 before doing problem 8.
• Select CC solutions
• Video (parts 1, 2, and 3) and 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:

• Your proof of 10(b) will likely not use the fact that the matrix is in reduced row echelon form (ie, your proof will probably apply for any matrix that’s just in row echelon form). It’s 10(c) where you’ll need to use the fact that the matrix is in reduced row echelon form.
• In the 2015 printing of the textbook, there’s a small typo in problem 14. The formula for $$h$$ given in the book has an $$x_j$$ instead of $$x$$. In other words, the correct formula is $h(x) = \sum_{i = 1}^n b_i \prod_{j \neq i} \frac{x - a_j}{a_i - a_j} \in k[x].$ This typo was corrected in the 2018 printing.
• The hint for problem 10(b) should say $$g_j = x_\ell + D$$ and $$S(g_i, g_j) = x_\ell C - x_s D$$. This typo was corrected in the 2018 printing.
• Video (parts 1 and 2) and 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:

• This reading is optional. That being said, note that some material in later sections (eg, sections 3.5 and 4.1) will only be accessible if you at least have a surface understanding of some concepts from this section.

## 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:

• For problem 10(a), it might be helpful to notice that the statement of the extension theorem does not require that you start with a Gröbner basis. Of course, it’s usually useful to apply the statement with a Gröbner basis with respect to lex order (since that helps us identify what the elimination ideal actually is), but the statement of the theorem applies more generally, and that observation will probably be useful for problem 10(a).
• For problem 14(a), you don’t really need to have done problem 13. Just use the statement of the rational implicitization theorem.
• For problem 12(b), $$W = V(uv)$$, and the ideal $$J$$ is the one defined on the bottom of page 139.
• Select CC solutions
• Video (“Zariski closure”) and notes
• Video (“Implicitization,” parts 1, 2, and 3) and notes

## 3.4: Singular Points on Curves

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

Notes:

• Feel free to skip the bit about envelopes. None of the exercises above are about envelopes.
• This reading could probably have been a part of chapter 1. You won’t need any Gröbner bases for the exercises about singular points.

## 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:

• This section is a challenging read. It also uses stuff about “standard representations” from section 2.9 (which was also an optional reading). If you decide to read it, I encourage you to read it twice. Read it once, skipping over the proof of theorem 2 (instead, focus on the statement of the theorem and the example right after the proof). Then work on some exercises, and then go back and try reading the proof of theorem 2.
• Problem 4(b) should say $$g_o = g_2$$. (The 2015 printing says $$g_o = g_3$$, but this was corrected in the 2018 printing.)

## 4.1: Nullstellensatz

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

Notes:

• The proof of the Weak Nullstellensatz uses some material from section 2.9. If you didn’t read section 2.9, feel free to skip this proof, but do make sure you understand the statement of the Weak Nullstellensatz!
• Video and 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:

• You should look through Appendix A as well.
• The discussion towards the end of the section about the radical of principal ideal generalizes problems 1.5.12–14.
• You might find problem 2.7.6 useful for problem 17 above.