Ideals, Varieties and Algorithms
Math 473 - Fall 2018
Announcements
The semester is over.
About the course
The aim of this course is to introduce students to the basic principles of algebraic geometry. Students are encouraged to use computer tools such as Pari GP, Magma, Desmos and SageMath to investigate examples.
We will explore the geometry of certain mathematical objects, named varieties, by using the algebraic structure of polynomial rings. A significant part of this study will be the theory of Gröbner basis.
At the end of the course students will be able to answer questions like: Does a given system of polynomials have finitely many solutions? If so what are they? If there are infinitely many solutions, how can these be described and understood?
Schedule of Lectures and Office Hours
Lectures: Tuesday 09:40-10:30 in M102 and Thursday 10:40-12:30 in M231.
Attendance will not be taken during the lectures. However, I strongly suggest that you shall attend the lectures regularly.
Office hours: Tuesday between 10:40-12:30 in M141.
If these office hours are not suitable for you, then you shall send me an email to fix an appointment. You may also stop by my office. If I am suitable, I will be happy to answer your questions.
Textbooks and Course Outline
The textbook is Cox, Little and O'Shea - Ideals, Varieties and Algorithms (4th edition). Its third edition is available in the reserve part of the library with code QA564.C688.
You can find the course outline below. For each week, we will cover the content of the indicated pages of our textbook.
(Oct 02, Oct 04) Polynomials and Affine Space, Affine Varieties 1--13.
(Oct 09, Oct 11) Parametrizations of Affine Varieties, Ideals 14--32.
(Oct 16, Oct 18) Ideals, Polynomials of One Variable 33--48.
(Oct 23, Oct 25) Introduction (Gröbner Bases), Ordering on the Monomials 49--60.
(Oct 30, Nov 01) A (multivariable) Division Algorithm 61--69. Monomial Ideals 70--72
(Nov 06, Nov 08) Dickson's Lemma 72--75. The Hilbert Basis Thm and Gröbner Bases 76-82
(Nov 13, Nov 15) Properties of Gröbner Bases 83--89.
(Nov 20, Nov 22) Buchberger's Algorithm 90--96.
(Nov 27, Nov 29) Applications and Refinements 97--108.
(Dec 04, Dec 06) Improvements on Buchberger's Algorithm 109--120.
(Dec 11, Dec 13) Elimination and Extension, The Geometry of Elimination 121--132.
(Dec 18, Dec 20) Implicitization, Singular Points and Envelopes 133--154.
(Dec 25, Dec 27) Gröbner Bases, Resultants and the Extension Theorem 155--174.
(Jan 01, Jan 03) An introduction to algebraic geometry.
Suggested Problems
Chap 1, Sec 1: 5, 6.
Chap 1, Sec 2: 1, 2, 4, 6, 7, 8, 9, 10.
Chap 1, Sec 3: 1, 2, 3, 4, 6, 7, 11.
Chap 1, Sec 4: 3, 6, 7, 8, 10, 11, 12, 16, 17.
Chap 1, Sec 5: 3, 4, 6, 8, 9, 10, 12, 17.
Chap 2, Sec 1: 1, 3, 4.
Chap 2, Sec 2: 1, 3, 4, 7, 8, 10, 11.
Chap 2, Sec 3: 1, 2, 3, 6, 7, 9.
Chap 2, Sec 4: 1, 3, 4, 5, 7, 8, 9.
Chap 2, Sec 5: 1, 2, 7, 8, 10, 11, 16, 17.
Chap 2, Sec 6: 1, 3, 4, 5, 6, 10.
Chap 2, Sec 7: 2, 3, 5, 7, 9, 11.
Chap 2, Sec 8: 1, 3, 5, 8, 10, 11.
Chap 2, Sec 9: 1, 2.
Chap 3, Sec 1: 2, 3, 4, 7, 9.
Chap 3, Sec 2: 3, 4, 5.
Chap 3, Sec 3: 6, 7, 8, 9, 12, 13, 14.
Chap 3, Sec 4: 1, 3, 5, 7, 9, 10, 11, 12, 13, 14, 16, 17.
Chap 3, Sec 5: 1, 2, 4.
Chap 3, Sec 6: 1, 2, 3, 4, 5, 6, 9, 12, 13, 15.
Exams and Grading
Your final letter grade will be determined by two midterms and the final exam. The exam dates are as below.
Midterm 1 - November 7 (%30)
Midterm 2 - December 13 (%30)
Final - January 18 (%40)
The envelope example with Desmos
In the following example of our textbook, we consider a family of circles of radius 2 centered at the points on the parabola y=x^2. This is an example of an envelope in which singular points play an interesting role. Click here to see how this animation is implemented.
Click here to see another example where the parabola is changed with y=x^3-x.
Sagemath Parametric Plot
Click here for Sagecell. Plug in the following code and submit.
u=var('u')
v=var('v')
parametric_plot3d( (u^2/v,v^2/u,u),(u,-3,3),(v,-3,3))
Magma Calculator
Click here for the Magma Calculator. Plug in the following code and submit.
P<x,y> := PolynomialRing( RationalField() , 2 , "glex" );
I := ideal< P | x^3-2*x*y , x^2*y-2*y^2+x >;
GroebnerBasis(I);
Pari Scripts
Pari GP is a free software by which you can create mathematical algorithms in a simple way with the help of built in functions.
Univarite: m473-oct18.gp
Lexicographic: m473-oct25.gp
Multivariable Division Algorithm: m473-nov01.gp
Buchberger's Algorithm: m473-nov22.gp
Minimal Grobner Basis: m473-nov29.gp
You can download Pari GP from here. It is also avaiable for Android.
Make-up and NA policy:
Only one make-up examination will be offered. The excuse for not attending an examination must be proved with documents. The make-up examination will take place shortly after the final exam. If you have missed both midterms, your letter grade would be NA. If you receive NA, then you can not take the final exam and the makeup.