MATH 5330 HOMEWORK (Fall 2016)

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Book = Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics, by David A. Cox, John Little, Donal O'Shea, Springer, 4th edition; denoted [CLO] in this course.
Note that there might be more than one correct answer for any given question.

LAST REVISION: 12/06/16

You should make sure you are viewing the most current version of this page and not the version in your browser's cache; reload the page from source. Dates indicate homework assigned in lecture that day. Dates for future assignments are tentative, and subject to change.
Skimming through the main ideas in a section shortly before that section is covered in class should help you understand the lecture -- try it!

Also, see the list of other books that might be helpful to your learning of this material. 

Aug 25
 
Attendance will be recorded, starting today.
Read course syllabus carefully. Review course website and repeat frequently during the semester.  
Read  https://vancliff.utasites.cloud/T/S09/1426study.pdf   and
read    https://www.jeffreybennett.com/pdf/How_to_Succeed_general.pdf  for ideas on how to study most effectively.

Read lecture notes & Sec 1.1 & do: 5, 3(d), 2.  Due Sept 22 in class.
 
Aug 30
 
Read lecture notes & Sec 1.2 & do: 1, 3, 4(a)-(d), 5, H1, 6, 8, 15(a)-(c).  Due Sept 22 in class.
Be careful to explain your answer to the parts of the questions that ask if the dimension matches what your 
intuition says it should be.
 
Sep 01
 
Read lecture notes & pgs 14-20 & do Sec 1.3: 1-4.  Due Sept 22 in class.
 
Sep 06
 
Read lecture notes & Sec 1.4 & do Sec 1.4: 1, 2, 3 (with ℚ replaced by k & assume ℕ⊂ k ), 5, H2, 7, 8,
15(a)(b), H3.  Due Sept 22 in class.
 
Sep 08
 
Read lecture notes and skim Sec 1.5.  Do Sec 1.5: 1, 5.  Due Oct 6 in class.
 
Sep 13
 
Read lecture notes and the statement of Theorem 4 on page 77 and pages 80-81.
Do Sec 2.5: 17, 18, 11, H4, H5, 16, H6-H8, 13, 14.  Due Oct 6 in class.
 
Sep 15
 
Read lecture notes & pg 209.7(Defn 7)-211 & do H9.  Due Oct 6 in class.
 
Sep 20
 
Read lecture notes & pgs 189-192 & do H10.   Due Oct 6 in class.
 
Sep 22
 
Read lecture notes.
 
Sep 27
 
Read lecture notes & do H11-H14 & do Sec 4.3: 3 (hint: read Definition 12 & Proposition 13 of Sec 4.3).
Due Oct 20 in class.
 
Sep 29
 
Read lecture notes & Definition 3, Theorem 4 & Theorem 6 of Sec 4.6.
Do Sec 4.6: 9.  Due Oct 20 in class.
 
Oct 04
 
Read lecture notes & do H15-H19.   Due Oct 20 in class.
 
Oct 06
 
Read lecture notes & do Sec 4.8: 4.  Note that this exercise is on pg 231 and appears to have an erroneous 
section number at the start of those exercises.    Due Oct 20 in class.
 
Oct 11
 
Read lecture notes & do Sec 4.4: 1 (this can be done in an informal manner).  
Due Nov 10 in class.
 
Oct 13
 
Read lecture notes & Sec 5.1 & do Sec 5.1: 1, 2, 5, 6.  Due Nov 10 in class.
 
Oct 18
 
Read lecture notes & Sec 5.2 (but note that Definition 8(iii) is not correct on pg 243 – what is wrong with it?).
Do Sec 5.2: 6, 9(use FIT), 10(a), 18.  Due Nov 10 in class.
 
Oct 20
 
Read lecture notes & do H20.  Read lecture notes & Sec 5.4 & do: 4 (assume V ⊆ A3), 5 (assume f, g ∊ k[x, y, z]) 
&14 (14 is optional for undergraduate students).  For 14(a), ignore the instruction to use material developed earlier; instead 
 use the fact that the field is ℝ so that 5th roots of real numbers exist.  For 14(c): use y5 = x2 in  ℝ[V] to show that g5 = f2 in ℝ[t]; then 
 deduce  f = h5, g = h2 for some h ∊ ℝ[t]; & then use (b) and the uniqueness part of (a) to prove that y = Φ(h)2 = (a + bx)2, where
a, b ∊ ℝ[y], yields a contradiction.  Due Nov 10 in class.
 
Oct 25
 
Read lecture notes.
 
Oct 27
 
Read lecture notes & Sec 5.5 & do  H21,  H22.  Due Nov 22 in class.
(Note that H21 should address a question that was asked by Mohammad in class.)
 
Nov 01
 
Read lecture notes.
 
Nov 03
 
Read lecture notes.
 
Nov 08
 
Read lecture notes & Sec 8.1 & do: H23-H25.   Due Nov 22 in class.
 
Nov 10
 
Read lecture notes & Sec 8.2 & do: 11, 12, 21(a).  Due Nov 22 in class.
 
Nov 15
 
Read lecture notes & do Sec 8.2: 7 (note that Proposition 5 in Sec 1.1 should be useful in (b)).
This exercise completes the part of the proof of Proposition 4 in Section 8.3 that was not proved in class.
Due Dec 1 in class.
 
Nov 17
 
Read lecture notes & do H26.   Due Dec 1 in class.
 
Nov 22
 
Read lecture notes & do Sec 8.4: 9 (hint: use the fact that the projective closure of W is the smallest projective
variety that contains W).
Do H27, H28.  Due Dec 1 in class.
 
Nov 24
 
THANKSGIVING  HOLIDAY – no lecture.
 
Nov 29
 
Read lecture notes & do H29, H30.  Due Dec 1 in class.
 
Dec 01
 
Read lecture notes and pages 451-454.
 
Dec 06
 
Read lecture notes.
Remember to complete online the student feedback survey by 11 pm on Wed Dec 7 -- check your mymav e-mail for the link.
I appreciate the feedback (e.g., quality of website?); thank you!
 
Dec 10
 
Approx 2:15-4:15 pm:  office hours in PKH 462
 
Dec 12
 
Approx 2:30-4:00 pm:  office hours in PKH 462
 


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