IDEALS AND VARIETIES

Course Code:

MATH 426

Course Type:

Area Elective

P:

3

Lab:

0

Laboratuvar Saati:

0

Credits:

3

ECTS:

6

Prerequisite Courses:

ABSTRACT ALGEBRA

Course Language:

English

Course Objectives:

The aim of this course is to introduce students to the preliminary concepts for classical algebraic geometry and understand the links between geometry and algebra.

Course Content:

Polynomial rings, ideals and varieties. Monomial orderings. Monomial ideals and Dickson's lemma. The Hilbert Basis Theorem and Gröbner Bases. Properties of Groebner bases. Buchberger's algorithm. Applications of Groebner bases. Elimination and Extension theorems. Resultants and the extension theorem.

Course Methodology:

1: Lecture, 2: Problem Solving

Course Evaluation Methods:

A: Written examination, B: Homework

Vertical Tabs

Course Learning Outcomes

Learning Outcomes	Program Learning Outcomes	Teaching Methods	Assessment Methods
1) Learn ideals of polynomial rings and varieties.	1,2,3	1	A,B
2) Learn monomial ordering and monomial ideals.	1,2,3	1	A,B
3) Learn Hilbert Basis Theorem and Gröbner Bases	1,2,3	1	A,B
4) Understand Buchberger's algorithm.	1,2,3	1	A,B
5) Understand the link between basic geometric shapes and polynomial ideals.	1,2,3	1	A,B

Course Flow

Week	Topics	Study Materials
1	Polynomials and Affine Space
2	Affine Varieties
3	Parametrizations of Affine Varieties
4	Ideals
5	Polynomials of One Variable
6	Orderings on the Monomials in k[x1,..., xn].
7	A Division Algorithm in k[x1,..., xn]
8	Monomial Ideals and Dickson’s Lemma
9	he Hilbert Basis Theorem and Gröbner Bases
10	Properties of Gröbner Bases, Buchberger’s Algorithm
11	First Applications of Gröbner Bases
12	Refinements of the Buchberger Criterion
13	Improvements on Buchberger’s Algorithm
14	Elimination Theory, The Elimination and Extension Theorems

Recommended Sources

Textbook	Cox, Little and O'Shea - Ideals, Varieties and Algorithms
Additional Resources

Material Sharing

Documents
Assignments
Exams

Assessment

IN-TERM STUDIES	NUMBER	PERCENTAGE
Mid-terms	1	60
Quizzes		-
Assignments	3	40
Total		100
CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE		40
CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE		60
Total		100

Course’s Contribution to Program

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamental research fields in mathematics (i.e., analysis, algebra, differential equations and geometry)				x
2	Acquiring fundamental knowledge on fundamental research fields in mathematics					x
3	Ability form and interpret the relations between research topics in mathematics					x
4	Ability to define, formulate and solve mathematical problems					x
5	Consciousness of professional ethics and responsibility			x
6	Ability to communicate actively			x
7	Ability of self-development in fields of interest				x
8	Ability to learn, choose and use necessary information technologies			x
9	Lifelong education			x

ECTS

-

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