MATH 5330 HOMEWORK (Fall 2020)

MATH 5330 HOMEWORK (Fall 2020)

Back to Dr Vancliff's home page Back to Dr Vancliff's Fall 2020 teaching page

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.

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 or clear the cache and reload.
There might be more than one correct answer for any given question.
Dates indicate homework assigned in lecture on that date. Dates for future assignments are tentative, and subject to change. (However, it will be similar to the homework posted here.)
On some dates, the questions are in a nonincreasing order; that is the order recommended by your instructor and is usually chosen to match the order in which the material was presented in class.
Any optional exercises are for those students who wish to explore some of the ideas in more depth.
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 (that file is from Fall 2016’s course, but is relevant to this course too).
LAST REVISION: 12/04/20.

Aug 29	Attendance will be noted, starting today. Read course syllabus carefully. Review course website and repeat frequently during the semester. Review the Canvas course website. Read https://vancliff.utasites.cloud/T/study.pdf and read https://www.jeffreybennett.com/pdf/How_to_Succeed_general.pdf for ideas on how to study most effectively.
Aug 29	Read lecture notes & Sec 1.1 & do Sec 1.1: 2, 3(d), 5. Due Sept 14 via Canvas.
Sep 05	Read lecture notes & Sec 1.2 & do Sec 1.2: 1, 3, 4(a)-(d), 5, H1, 6, 8, 15(a)-(c). 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; solutions with missing explanations will be returned ungraded. Due Sept 14 via Canvas.
Sep 05	Read lecture notes & pgs 14-20 & do Sec 1.3: 1-4. Due Sept 14 via Canvas.
Sep 12	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. Read lecture notes and skim Sec 1.5. Do Sec 1.5: 1, 5. Due Sept 30 via Canvas.
Sep 12	Read lecture notes and the statement of Theorem 4 on page 77. Watch the video in Canvas under Modules entitled “Part 3 of Sept 12 Lecture”. Do Sec 2.5: 17, 18, 11, H4, H5, 16. Due Sept 30 via Canvas.
Sep 19	Read lecture notes and do H6-H9. (Note that H8 was revised on Sept 26.) Due Sept 30 via Canvas.
Sep 19	Read lecture notes and pages 80-81. Do Sec 2.5:13, 14. Due Sept 30 via Canvas.
Sep 26	Read lecture notes & pg 209.7(Defn 7)-210 & pgs 189-190 & do H10 & Sec 4.3: 3 (hint: read Definition 12 through Proposition 13 of Sec 4.3). Due Oct 15 via Canvas.
Sep 26	Read lecture notes & do H 11. Due Oct 15 via Canvas.
Oct 03	Read lecture notes & pgs 190-192 and do H12-H 14. Due Oct 15 via Canvas.
Oct 03	Read lecture notes & watch the video in Canvas under Modules entitled “Part 3 of Oct 3 Lecture”. Do H15 and read Definition 3, Theorem 4 & Theorem 6 of Sec 4.6 and do Sec 4.6: 9. Due Oct 15 via Canvas.
Oct 10	Read lecture notes.
Oct 10	Read lecture notes & watch the video in Canvas under Modules entitled “Part 3 of Oct 10 Lecture”. Do H16-H19 and Sec 4.8: 4. Note that this latter exercise is on pg 231 and appears to have an erroneous section number at the start of those exercises. Due Oct 30 via Canvas.
Oct 17	Read lecture notes & do Sec 4.4: 1 (this last question can be done in an informal manner). Due Oct 30 via Canvas.
Oct 17	Read lecture notes & Sec 5.1 & Sec 5.2 (but note that Definition 8(iii) is not correct on pg 243 – what is wrong with it?). Do Sec 5.1: 1, 2, 5, 6 & Sec 5.2: 6, 9(use FIT), 10(a), 18, H20(a)-(f). Due Oct 30 via Canvas.
Oct 24	Read lecture notes.
Oct 24	Read lecture notes & Sec 5.4 & watch the video in Canvas under Modules entitled “Part 3 & Part 4 of Oct 24 Lecture”. Do Sec 5.4: 4 (assume V ⊆ 𝕜³), 5 (assume f, g ∊ 𝕜[x, y, z]) & 14 (14 is optional for any 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 = x²in 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 13 via Canvas.
Oct 31	Read lecture notes & Sec 5.5 & watch (optional) the video (Part 3) in Canvas under Modules entitled “Part 3 & Part 4 of Oct 31 Lecture” & do H21, H22. (Note that the additional homework should be dated Oct 31, 2020, or later – you might need to refresh the page or delete the cache to access the most recent version of that file. Check the date on page 1 of that file.) Due Nov 13 via Canvas.
Oct 31	Read lecture notes & Sec 8.1 & watch the video (Part 4) in Canvas under Modules entitled “Part 3 & Part 4 of Oct 31 Lecture” & do: H23. Due Nov 13 via Canvas.
Nov 07	Read lecture notes & Sec 8.2 & do: H2 4-H25. Due Nov 30 via Canvas.
Nov 07	Read lecture notes & watch the video in Canvas under Modules entitled “Part 3 of Nov 7 Lecture” & do Sec 8.2: 11, 12, 21(a), 7 (note that Proposition 5 in Sec 1.1 should be useful in (b)). The last exercise completes the part of the proof of Proposition 4 in Section 8.3 that was not proved completely in lecture or in the extra video. Due Nov 30 via Canvas.
Nov 14	Read lecture notes & do H26. Due Nov 30 via Canvas.
Nov 14	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). Due Nov 30 via Canvas.
Nov 20	Read lecture notes & watch the video in Canvas under Modules entitled “Part 3 of Nov 20 Lecture” & do H27, H28. Due Dec 8 via Canvas.
Nov 20	Read lecture notes & do H29, H30. Due Dec 8 via Canvas.
Nov 28	THANKSGIVING HOLIDAY – no lecture.
Nov 28	THANKSGIVING HOLIDAY – no lecture.
Dec 05	Read lecture notes and pages 451-454.
Dec 05	Remember to complete online the student feedback survey by 11 pm on Tues Dec 8 -- check your mymav e-mail for the link. I appreciate the feedback (e.g., quality of website, quality of videos, quality of conference lectures, use of ipad for lectures, use of 2 video feeds for lectures, use of Canvas); thank you! Office hours through Tues Dec 8 are as usual. For office hours after Dec 8, see below…..
Dec 10	12:00-1:00 PM office hour
Dec 14	1:00-2:00 PM office hour
Dec 16	1:00-2:00 PM office hour

Back to Dr Vancliff's Fall 2020 teaching page

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 or clear the cache and reload.